A System of Logic

JOHN STUART MILL

(1806-1873)

Contents

HOW TO READ THE TEXT

Introduction

Book I: Names and propositions

Glossary

Chapter 1. The need to start with an analysis of language

Chapter 2. Names

Chapter 3. Things that are denoted by names

Chapter 4. Propositions

Chapter 5. The import of propositions

Chapter 6. Merely verbal propositions

Chapter 7. The nature of classification. The five predicables.

Chapter 8. Definition

Book II: Reasoning

Glossary

Chapter 1. Inference of reasoning in general

Chapter 2. Reasoning, or syllogism

Chapter 3. The functions and logical value of the syllogism

Chapter 4. Trains of reasoning, and deductive sciences

Chapter 5. Demonstration, and necessary truths

Chapter 6. Demonstration, and necessary truths (cont'd)

Chapter 7. Examining some of the opposition to the preceding doctrines

Book III: Induction

Glossary

Chapter 1. Preliminary remarks about induction in general

Chapter 2. Inductions improperly so called

Chapter 3. The ground of induction

Chapter 4. Laws of nature

Chapter 5. The law of universal causation

Chapter 6. The composition of causes

Chapter 7. Observation and experiment

Chapter 8. The four methods of experimental inquiry

Chapter 9. Examples of the four methods

Chapter 10. Plurality of causes, and the intermixture of effects

Chapter 11. The deductive method

Chapter 12. Explaining laws of nature

Chapter 13. Examples of the explanation of laws of nature

Chapter 14. The limits to the explanation of laws of nature. Hypotheses

Chapter 15. Progressive effects. The continued action of causes

Chapter 16. Empirical laws

Chapter 17. Chance and its elimination

Chapter 18. The calculation of chances

Chapter 19. Extending derivative laws to adjacent cases

Chapter 20. Analogy

Chapter 21. Evidence for the law of universal causation

Chapter 22. Uniformities of coexistence that don’t depend on causation

Chapter 23. Approximate generalisations. Probable evidence

Chapter 24. The remaining laws of nature

Chapter 25. The grounds of disbelief

Book IV: Operations Subsidiary to Induction

Glossary

Chapter 1. Observation and description

Chapter 2. Abstraction or the formation of conceptions

Chapter 3. Naming, as subsidiary to induction

Chapter 4. What is needed for a philosophical language. The principles of definition

Chapter 5. The natural history of the variations in the meaning of terms

Chapter 6. The principles of a philosophical language further considered

Chapter 7. Classification, as subsidiary to induction

Chapter 8. Classification by series

© Jonathan Bennett 2017

A System of Logic

Ratiocinative and Inductive

Presenting a Connected View of the Principles of Evidence and the Methods of Scientific Investigation

JOHN STUART MILL

1843

Early Modern Texts, 2017

COPYRIGHT

A System of Logic

© Jonathan Bennett, 2017. All rights reserved.

Early Modern Texts

First launched: April 2013

Contact: earlymodern31@gmail.com

www.earlymoderntexts.com

HOW TO READ THE TEXT

[Brackets] enclose editorial explanations. Small ·dots· enclose material that has been added, but can be read as though it were part of the original text. Occasional •bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. Every four-point ellipsis . . . . indicates the omission of a brief passage that seems to present more difficulty than it is worth. In this work such omissions are usually of unneeded further examples or rewordings. Longer omissions are reported between brackets in normal-sized type. Three-point ellipses . . . indicate omissions by Mill in passages he quotes from others.

When a word is spoken about in this version, it is usually put between quotation marks; Mill himself does that with phrases and sentences but not with single words.

Mill here refers to contemporaries by their surnames; in the original he is less abrupt—‘Archbishop Whateley’, ‘Dr Whewell’, ‘Professor Bain’, and so on.

Introduction

§1. Writers on logic have differed as much in their definitions of it as in their handling of its details. This isn’t surprising in a subject where writers have used the same language to convey different ideas. It’s the same in ethics and jurisprudence. Almost every writer, having his own views about some aspects of these branches of knowledge, has framed his definition in a way that shows beforehand what his particular views are, sometimes begging the question [see Glossary] in their favour.

This diversity is an inevitable result—and to some extent a proper result—of the imperfect state of those sciences [see Glossary]. You can’t expect people to agree about the definition of anything if they don’t agree about the thing itself. To define something is to select from its properties the ones that are to be understood to be declared by its name; and we aren’t competent to make that selection until the thing’s properties are well known to us. And when the ‘thing’ is as complex as a science, the definition we start with is seldom one that we’ll still think appropriate when we know more. Until we know the details we can’t pick the most correct and compact way of gathering them under a general description. A reasonable definition of chemistry became possible only after men had acquired extensive knowledge of the details of chemical phenomena; and the definition of ·biology·—the science of life and organisation—is still a matter of dispute. . . . The definition that I’m going to give of the science of logic claims only to be a statement of the question that I have put to myself and that this book is an attempt to answer. You may object to it as a definition of logic, but it’s a correct definition of the subject of this volume.

§2. Logic has often been called ‘the art of reasoning’. Archbishop Whately, the writer who has done most to restore logic to the level of esteem which it used to have from educated people in England, has defined logic as the science and the art of reasoning; meaning

•by ‘science’: the analysis of the mental process that occurs when we reason, and

•by ‘art’ [see Glossary]: the rules, based on that analysis, for conducting the process correctly.

He was certainly right to add ‘the science and’: a system of rules governing the process must be based on a grasp of the mental process itself—the steps it consists of and the conditions it depends on. Art requires knowledge; art that has grown beyond its infant state requires scientific knowledge. Not every art bears the name of a science, but that’s because in many cases a single art is based on several sciences. . . .

So logic is the science and art of reasoning. But the word ‘reasoning’—like most scientific terms in popular [see Glossary] use—is highly ambiguous. In one of its senses it means syllogising, i.e. the type of inference in which we draw particular conclusions from general premises. . . . In another of its senses, to ‘reason’ is simply to infer any assertion [see Glossary] from assertions already accepted; and in this sense induction has as much right to be called ‘reasoning’ as have the demonstrations of geometry.

Writers on logic have generally preferred the former sense of the word, but I shall use it in the latter more extensive meaning. Every author has the right to define his subject, provisionally, in whatever way he pleases; but I think you’ll come to see in the course of this work that this should be not only the •provisional but also the •final definition. . . . It happens also to be the one that fits better with general usage of the English language.

§3. But ‘reasoning’—even in its widest permissible sense— seems to be too narrow to cover the whole of logic, according to •the best conception of logic’s scope, or even according to most current conception of this. The use of ‘logic’ to refer to the theory of argumentation comes from the Aristotelian logicians (the ‘scholastics’ as they are commonly termed). Yet even their textbooks present •Argumentation only in Part III, with Part I treating •Terms and Part II treating •propositions; and in one or other of these two Parts they included •Definition and •Division [see Glossary]. Some writers said that they were dealing with these previous topics only because of their connection with reasoning, and as a preparation for the doctrine and rules of the syllogism. But they treated them in much more detail. . . .than was required for that purpose. More recent writers on logic have generally understood the term ‘Logic’. . . .as equivalent to ‘the art of thinking’; and this sense of it isn’t confined to books and scientific inquiries. Even in ordinary conversation, the ideas connected with the word ‘logic’ include at least •precision of language and •accuracy of classification; and we probably hear ‘a logical arrangement’ or ‘logically defined’ more often than ‘logically deduced from the premises’. And a man is said to have ‘powerful logic’ not because of the accuracy of his deductions because

•of the extent of his command over premises; because

•he quickly comes up with many general propositions he needs to explain a difficulty or expose a fallacy;

because, in short,

•his knowledge, besides being ample, is well under his command for argumentative use.

So ordinary usage as well as the practice of experts support the inclusion in Logic of various intellectual operations other than reasoning and argumentation.

These various operations could be included in Logic in a very simple definition of logic as the science that deals with the operations of the human understanding in the pursuit of truth. For all the operations that logic has ever claimed to govern—naming, classification, definition etc.—can all be regarded as contrivances for enabling a person to know the truths he needs and to know them at the precise moment when he needs them. Those operations also serve other purposes, such as imparting our knowledge to others; but that doesn’t put them into the domain of Logic. Logic is concerned only with the guidance of one’s own thoughts: communicating them to others belongs to •Rhetoric in the broad sense the ancients gave to that term, or to the still more extensive art of •Education. Logic doesn’t concern itself with such inter-personal matters. If there were only one thinking being in the universe, he might be a perfect logician. . . .

§4. Whereas the definition of Logic in terms of argumentation includes too little, the definition in terms of truthseeking includes too much.

Truths are known to us in two ways—•directly, by themselves, and •through the medium of other truths. The former are the subject of intuition or consciousness;[1] the latter are the subject of inference. The truths known by intuition are the basic [see Glossary] premises from which everything else is inferred. Our assent to a conclusion is based on the truth of the premises; so we could never acquire knowledge by reasoning unless something could be known in advance of all reasoning.

We know by immediate consciousness our own bodily sensations and mental feelings. I know directly that I was angry yesterday and that I am hungry now. We know only through inference

(i) things that happened in our absence,

(ii) events recorded in history, and

(iii) the theorems of mathematics.

We infer (i) from testimony, (ii) from present traces of those past events, and (iii) from the premises laid down in books of geometry under the title of ‘definitions’ and ‘axioms’. Anything we can know must belong among the basic data or among the conclusions that can be drawn from these.

Logic as I understand it has nothing directly to say in answer to questions about the basic data or ultimate premises of our knowledge—

•How many are there?

•What are they like?

•How are they obtained?

•What tests are there to determine whether something is ultimate?

Some of the answers to these concern sciences other than logic; others fall outside the range of any science.

We can’t question anything that we know by consciousness. When we see or feel something—whether bodily or mentally—we can’t help being sure that we see or feel it. No •science is needed to establish such truths; no rules of •art can make our knowledge of them more certain. There is no logic for this part of our knowledge.

But we may imagine that we are seeing or feeling when really we are inferring. Something that results from a very rapid inference may seem to be learned intuitively. It has long been accepted by thinkers of the most opposite schools that we make this mistake in the familiar business of eyesight. We appear to ourselves to be absolutely directly conscious of an object’s distance from us. Yet it was discovered long ago

•that what is perceived by the eye is merely a variously coloured surface;

•that when we imagine we see •distance all we really see are •certain variations of apparent size and colour;

•that our estimate of how far away the object is results partly (i) from a rapid inference from the muscular sensations we get from focussing our eyes on the object and partly (ii) from a comparison (made too fast for us to be aware of making it) between the size and colour of the objects as they appear now and the size and colour of the same or of similar objects as they appeared when close at hand or when their distance from us was known by other evidence.

The perception of distance by the eye, which seems so like intuition, is thus really an inference based on experience—an inference that we learn to make, getting better at it as our experience increases. . . .

A scientific study of how the human understanding goes about the pursuit of truth includes the question: Which facts are objects of intuition or consciousness, and which are merely inferred? But this has never been considered a part of logic. It belongs in another quite different department of science known as ‘metaphysics’. That part of mental philosophy tries to discover what part of the mind’s furniture •belongs to it basically and what part •is constructed out of materials that come to it from outside. This science tackles questions about

•the existence of matter;

•the existence of spirits;

•the distinction between spirit and matter;

•the reality of time and space, as external to the mind and different from the things that are said to exist ‘in’ them.

These days most people accept that the existence of matter or of spirit, of space or of time, is intrinsically incapable of being proved, and that anything known of them must come from immediate intuition. Metaphysics also include the inquiries into the nature of conception, perception, memory, and belief, which are all operations of the understanding in the pursuit of truth; but the logician as such isn’t interested in them. Metaphysics also includes questions like these:

•To what extent are our intellectual faculties and our emotions innate? and to what extent do they result from association?

•Are God and duty realities whose existence is shown to us a priori by the constitution of our rational faculty? or are our ideas of them acquired notions, the origin of which we can trace and explain (so that their reality is to be settled not by consciousness or intuition but by evidence and reasoning)?

Logic deals only with the part of our knowledge that consists of inferences from previously known truths—general propositions or particular observations and perceptions. Logic is the science not of •belief but of •proof or evidence. Because belief professes to be based on proof, it is logic’s job to supply a test for determining whether a belief is well grounded. But it has nothing to do with the claim that this or that proposition has to belief on the evidence of consciousness—i.e. without evidence in the proper sense of the word,

§5. It’s generally agreed that most of our knowledge, whether of general truths or of particular facts, is reached by inference; so logic has authority over nearly all of science and of human conduct. . . . Everyone has daily, hourly, and momentary need to learn facts that he didn’t directly observe. . . ., because the facts are important to his interests or occupations. The whole business of the magistrate, the military commander, the navigator, the physician, the agriculturist, is to judge evidence and to act accordingly. They all have to establish certain facts so that they can then apply certain rules. . . .; and how well they do this will settle how well they do their jobs. Inferring is the only occupation that the mind is engaged in continuously. It is the subject not ·only· of logic but of knowledge in general.

Logic is not the same thing as knowledge, though the field of logic is co-extensive [see Glossary] with that of knowledge. Logic is the judge and evaluator of all particular investigations. Its role is not to •find evidence but to determine whether something that has been found is evidence. Logic doesn’t observe, invent, or discover—it judges. A surgeon/coroner wants to know whether this man died by violence; it’s not up to logic to tell him what the signs of that would be; he must learn that from his own experience or from that of other surgeons. But logic judges the sufficiency of that experience to justify his rules, and on the sufficiency of his rules to justify his conduct. It doesn’t give him proofs, but teaches him what makes them proofs, and how to judge them. It doesn’t teach that fact P proves fact Q, but says what conditions any fact must satisfy if it is to prove other facts. . . .

It is in this sense that logic is—as it has been called—the science of science itself. All science consists of data [see Glossary] and conclusions from them, of proofs and what they prove; and logic says how data must relate to anything that can be concluded from them. . . .

§6. . . . .A science can be developed to quite an advanced stage without using any logic except what thoughtful laymen pick up in the course of their studies. Mankind judged evidence, often correctly, before logic was a science; if they hadn’t, it could never have become a science. Similarly, they carried out great mechanical works before they understood the laws of mechanics. But there are limits to what engineers can achieve without principles of mechanics, and to what thinkers can do without principles of logic. There may be a few exceptions; but the bulk of mankind need either to •understand the theory of what they are doing, or to •have rules laid down for them by those who do understand it. In science’s progress from easier problems to harder ones, almost every big step has been preceded or accompanied by a corresponding improvement in the notions and principles of logic accepted by the most advanced thinkers. Some of the more difficult sciences are still very defective—with very little proved in them, and controversy about that little—and the reason for this is perhaps that men’s logical notions aren’t yet broad or accurate enough for the estimation of the evidence proper to those particular sciences.

§7. Logic, then, is the science of the mental activities that are involved in the estimation of evidence: both •the inferential move from known truths to unknown ones, and •all other intellectual operations that support this move. So it includes the operations of naming, defining and classifying. Why? Because language helps us to •think as well as to •communicate our thoughts. Quite apart from their role in communication, the operations of defining and classifying help us not only •to keep our evidences and the conclusions from them permanent and readily accessible in the memory but also •to organize the facts that we may at some time want to investigate, so as to enable us to perceive more clearly what evidence there is, and to judge more accurately whether it is sufficient. . . . Other more elementary processes are involved in all thinking—e.g. conception, memory, and so on—but there’s no need for logic to pay special attention to them, since they have no special connection with the problem of evidence. . . .

So I shall try •to conduct a correct analysis of reasoning, i.e. inference, and of whatever other mental operations as are intended to help reasoning, and also—along with this analysis and based upon it—•to collect or construct a set of rules or standards for testing the sufficiency of any given evidence to prove any given proposition.

In this analysis I shan’t try to decompose the mental operations in question into their ultimate elements. All that’s needed is for the analysis to be correct as far as it goes, and for it to go far enough for the practical purposes of the art of logic. If a proof doesn’t get the whole way from the premises to the conclusion, it achieves nothing; but an analysis can be valuable even if it doesn’t go the whole way down to the ultimate elements. Analytical chemistry’s results wouldn’t lose their value if it were discovered that all the supposedly ‘simple substances’ are really compounds. . . .

So I’ll try to analyse the process of inference (and processes that depend on it) only as far as may be needed to mark off correct from incorrect performances of those processes. . . . Logic’s opponents have said that we don’t learn to use our muscles by studying their anatomy. Actually, we might: if any of our muscles became weak or otherwise defective, this might be incurable without some knowledge of their anatomy. But if in a treatise on logic I pushed the analysis of reasoning beyond the point where any inaccuracy that has crept into it must become visible, I would be open to the criticism involved in this ·muscle-weakness· objection. The analysis of bodily movements should go far enough to enable us to distinguish movements that we ought to perform from the rest; and it’s the same with logic. . . . Logic has no interest in pushing the analysis beyond the point where it becomes apparent whether the operations in any individual case been rightly or wrongly performed. . . . The range of logic as a science is determined by its needs as an art: whatever it doesn’t need for its practical ends it leaves to ·metaphysics·, the larger science that corresponds not to any particular art but to all of them; it’s the science that deals with the constitution of the human faculties ·generally·, and it has the job of deciding which facts are ultimate and which can be further analysed into more basic facts. Few if any of the conclusions I’ll reach in this work are necessarily connected to any particular views about the further analysis. Logic is common ground on which the followers of Hartley and of Reid, of Locke and of Kant, can meet and join hands. They were all logicians as well as metaphysicians, so I may sometimes contradict things they say in the domain of logic; but the field on which their principal battles have been fought lies outside logic.

Logical principles aren’t altogether irrelevant to those more abstruse discussions; and our preferred solution to the problem that logic proposes is bound to favour one rather than another opinion on these controverted subjects. That is because metaphysics must use means whose validity is the business of logic. No doubt metaphysics does as much as it can by merely by attending more closely and intently to our •consciousness (or, more properly speaking, to our •memory); and logic doesn’t come into that. But when this method is insufficient to attain the end of its inquiries, metaphysics must like other sciences proceed by means of evidence; and the moment it begins to make inferences from evidence, logic stands in judgment over it. . . .

But this doesn’t relate logic more closely to metaphysics than it is related to every other science. I can conscientiously affirm that no one proposition laid down in this work has been adopted because of its favouring some opinion in any department of knowledge or of inquiry on which the theoreticians are still undecided.[2]

NOTES

[1] I use these terms interchangeably because for my present purposes there’s no need to distinguish them. Metaphysicians usually restrict ’intuition’ to the direct knowledge we’re supposed to have of things external to our minds, and ‘consciousness’ to our knowledge of our own mental phenomena.

[2] My view of the definition and purpose of logic stands in marked opposition to the position of a school of philosophy which is represented in England by the writings of Sir William Hamilton and of his numerous pupils. They see logic as ‘the science of the formal laws of thought’, a definition that they adopt so as to exclude from logic anything concerning belief and disbelief, or the pursuit of truth as such, thereby restricting it to one small area of its total province—namely the area having to do not with truth but with consistency. In my book An Examination of Sir William Hamilton’s Philosophy I have said all that I think it is useful to say against this limitation of the field of logic. . . .

Book I: Names and propositions

Glossary

Chapter 1. The need to start with an analysis of language

Chapter 2. Names

Chapter 3. Things that are denoted by names

Chapter 4. Propositions

Chapter 5. The import of propositions

Chapter 6. Merely verbal propositions

Chapter 7. The nature of classification. The five predicables.

Chapter 8. Definition

Glossary

accidentally: You have your height accidentally, meaning that your height could have been different without that affecting who you are.

art: In this work, ‘art’ is a vehicle for several related ideas: rules, skill, techniques.

assertion: Mill uses this in about the way we use ‘proposition’. For there to be an ‘assertion’, in his sense, no person needs to have asserted anything. Mill sometimes speaks of propositions as asserting this or that.

basic: This replaces Mill’s ‘original’ in some of its occurrences.

begging the question: Mill’s sense of this phrase is the only sense it had until fairly recently: ‘beg the question’ was to offer a ‘proof’ of P from premises that include P. It now means ‘raise the question’ (‘That begs the question of what he was doing on the roof in the first place.’) It seems that complacently illiterate journalists (of whom there are many) encountered the phrase, liked it, guessed at its meaning, and plunged ahead without checking.

cardinal: principal, most important, leading.

co-extensive: Used here in the sense that it still has: when Mill says (here) that ‘the field of logic is co-extensive with that of knowledge’ he means that any pursuit of knowledge will involve issues about logic, and that any study of logic will bring in issues about knowledge.

data: Mill’s readers will have understood ‘data’ as the plural of ‘datum’. Many years later it degenerated into a singular mass-term, like ‘soup’.

denote: In its root sense this mean ‘stand for’, ‘refer to’—so that ‘mankind’ denotes the human race, your name denotes you, and so does any description of the form ‘the. . . ’ that is true of you and nothing else. Here, however, we see Mill stretching the word in two ways: in ‘Abraham Lincoln was tall’, Mill would say that

•‘Abraham Lincoln’ denotes Abraham Lincoln;

•‘tall’ denotes tallness, and

•‘was’ denotes that something is being affirmed of something.

He doesn’t comment on the vast difference between ‘x denotes y’ and ‘x denotes that P’.

differentiae: Plural of ‘differentia’.

division: classification

frame: To frame an idea is to form it, cause it to exist in your mind; how you frame a proposition or definition has to do with how you shape it or formulate it. When Mill speaks of framing a class he means forming or creating a class.

identical proposition: Strictly speaking, this is a proposition of the form ‘x is x’, where the subject and predicate are identical. But the phrase came also to be used for any proposition where the meaning of the predicate is a part of the meaning of the subject.

import: In Mill’s use of it, this means about the same as ‘meaning’; but he does use both those words, and the present version will follow him in that.

induction: At the start of III.2 Mill defines this as ‘the operation of the mind by which we infer that what we know to be true in a particular case or cases will be true in all cases that resemble the former in certain assignable respects.’

meaning: In most places this is the word Mill has used, but sometimes it replaces his ‘acceptation’. It sometimes appears in the singular though the plural would seem more natural; that’s how Mill wrote it.

mutatis mutandis: A Latin phrase that is still in current use. It means ‘(mutatis) with changes made (mutandis) in the things that need to be changed’. The use of it implies that it’s obvious what the needed changes are.

name: Mill uses ‘name’ in such a way that ‘Bentham’ and ‘gold’ and ‘the author of Spies’ and ‘yellowness’ and ‘yellow’ are all names. The odd one out is ‘yellow’ but Mill insists that it names the same colour that ‘yellowness’ names. In the present version this usage of his will be strictly followed.

noumenon: A Greek word, much used by Kant, meaning ‘thing considered as it is in its own nature’ in contrast with ‘thing considered in terms of how it appears’, i.e. phenomenon. The plural is noumena. You’ll see here that Mill takes it for granted that noumena are the causes of phenomena.

popular: Even as late as Mill’s time this mainly meant ‘of the people’, usually the not highly educated or very intelligent people. It didn’t mean ‘liked by the people’.

principle: Mill nearly always uses ‘principle’ as you and I do, to stand for a special kind of proposition. But the word used to have a common meaning, now obsolete, in which ‘principle’ means ‘source’, ‘cause’, ‘driver’, ‘energizer’, or the like; and Mill uses it just twice, in the same paragraph (here, in the phrase ‘thinking principle’—meaning ‘whatever it is that drives our mental processes’.

proximate kind: The lowest kind—corresponding to the smallest class—in an Aristotelian classification. See here.

real: Here the word ‘real’ is tightly tied to its origin in the Latin res = ‘thing’. So the contrast between ‘real’ propositions and ‘verbal’ ones involves the contrast between things and words.

science: Any intellectual discipline whose doctrines are are highly organised into a logical structure. It doesn’t have to involve experiments, or to be empirical. Many philosophers thought that theology is a science.

signification: This seems to mean about the same as ‘meaning’, but Mill uses both words, and this version will respect his choices.

summa genera: The plural of summum genus = ‘highest class’. Mill explains this well enough on here.

Chapter 1. The need to start with an analysis of language

§1. Writers on logic often start their treatises with a few general remarks (usually rather meagre ones, admittedly) about terms and their varieties. ·I’ll be doing that too·, and perhaps you won’t require from me a detailed justification for thus •following common practice, as you would if I were to •deviate from it.

The reasons for it, indeed, are far too obvious to require a formal justification. Logic is a part of the art of thinking: and language is agreed by all philosophers to be obviously one of the principal instruments of thought; and any imperfection in the instrument, or in how it is used, is agreed to be liable. . . .to confuse and impede the process of thinking and destroy all ground of confidence in the result. For someone to try to study methods of philosophising before he has become familiar with the meaning [see Glossary] and right use of various kinds of words would be like someone trying to become an astronomical observer before he has learned to adjust the focal distance of his telescope!

Reasoning or inference, the principal subject of logic, is usually done by means of words, and in complicated cases it can’t be done in any other way; so anyone who doesn’t have a thorough insight into the signification [see Glossary] and purposes of words will almost certainly reason or infer incorrectly. Logicians have generally felt that unless they removed this source of error at the outset, their pupils wouldn’t be able to learn anything useful from them. . . . That’s why it has always been thought that the study of logic must start with as deep an inquiry into language as is needed to guard against the errors to which language gives rise.

But there’s a deeper reason why the logician should start by considering the import [see Glossary] of •words—namely that if he doesn’t start there he can’t examine the import of •propositions; and they stand right at the threshold of the science of logic.

In the Introduction I said that logic aims to discover •how we come by the part of our knowledge (much the biggest part) that isn’t intuitive; and •by what criterion we can distinguish what is proved from what isn’t, what is worthy of belief from what isn’t. . . . Logic is concerned with questions that can’t be answered from direct consciousness, but only on the basis of evidence. But we can’t inquire into how to answer questions until we have inquired into what questions there are—what inquiries are there that we might think could be answered? The best route to an answer to that is through a survey and analysis of propositions.

§2. The answer to any possible question must be contained in a proposition or assertion. Anything that can be believed— or even disbelieved—must when put into words have the form of a proposition. All truth and all error lie in propositions. When we speak of ‘a truth’, we mean ‘a true proposition’; and errors are false propositions. . . . The questions

•How many kinds of inquiries can be propounded?

•how many kinds of judgments can be made?

•how many meaningful kinds of propositions can be formulated?

are in fact merely different forms of a single question. So a good survey of propositions and of their varieties will tell us •what questions mankind have actually asked themselves and •what answers they have thought they had grounds to believe.

We can see at a glance that a proposition is formed by putting together two names [see Glossary]. According to the common definition (which is good enough for my purposes)

‘proposition’ = ‘discourse in which something is affirmed or denied of something’.

Thus, in ‘Gold is yellow’ the quality yellow is affirmed of the substance gold. . . .

Every proposition has three parts: the subject, the predicate, and the copula. The predicate is the name denoting x whatever it is that is affirmed or denied. The subject is the name denoting the person or thing of which x is affirmed or denied. The copula is the sign denoting that there is an affirmation or denial, thus enabling the hearer or reader to distinguish a proposition from any other kind of discourse. In ‘The earth is round’,

•the predicate is ‘round’, which denotes the quality that is affirmed or (as they say) predicated;

•the subject is ‘the earth’, which denotes the object of which that quality is affirmed; and

•the copula is ‘is’, which is a connecting mark between the subject and the predicate, showing that one is affirmed of the other.

Set the copula aside for the present; I’ll return to it later.

So we can say that every proposition consists of at least two names—brings two names together in a particular manner. This shows us that for an act of belief one object isn’t enough; the simplest act of belief has something to do with two objects—two names and (since the names must name something) two nameable things. Many thinkers would cut the matter short by saying ‘two ideas’. They would say that the subject and predicate are names of ideas, and that when someone believes that gold is yellow he is bringing one of these ideas ‘under’ the other (that’s how they often express it). We’re not yet in a position to evaluate this account of believing. At present we must settle for saying that in every act of belief two objects are in some way attended to—that anything that doesn’t embrace two distinct subjects of thought, whether material or intellectual, can’t be a belief or a question. Each of the subjects of thought may be conceived by itself or found to be inconceivable by itself, but there’s no question of its being believed by itself.

[Mill illustrates this with ‘the sun’: this is meaningful, and gives a direction to the hearer’s thought, but it can’t be true and can’t be believed. But if we move to ‘the sun exists’, which is] the assertion that involves the least reference to any object besides the sun, we now have something that a person can say he believes. And it involves two objects of conception—the sun, and existence. You may want to say that

the second conception is involved in the first, ·so that really there is only one object here, not two·;

but this is wrong, because the sun can be conceived as no longer existing. . . . Similarly, ‘my father’ doesn’t include all the meaning of ‘my father exists’, for he may be dead; ‘a round square’ doesn’t include the meaning of ‘a round square exists’, because it doesn’t and can’t exist. . . .

§3. That first step in the analysis of the object of belief seems obvious but it will turn out to be quite important. We can’t go further with that analysis until we have made a preliminary survey of language. If we tried to take more steps along that same path of analysing the import of •propositions we would find that we couldn’t do this until we had looked into the import of •names. . . . Now, what happens in our mind when we affirm or deny one name of another must depend on what they are names of, because our affirmation or denial isn’t about the mere names themselves, but about what they stand for. So we have here a new reason why the signification of names, and the relation between names and the things they signify, must be the next thing we inquire into.

Here is something that might be said:

The most we can get out of the meaning of names is a guide to the opinions—possibly foolish opinions—that mankind have formed concerning things. The object of philosophy is truth, not opinion; so the philosopher should dismiss words and look into things themselves to discover what questions can be asked and answered regarding them.

No-one could follow this advice. And, anyway, what it really does is to urge the philosopher to discard all the results of the labours of his predecessors, and behave as though he were the first person who had ever looked on nature with an inquiring eye! What does anyone’s personal knowledge of things amount to after subtracting everything he has learned through the words of other people? Even after he has learned as much as people usually do learn from others, will the notions of things contained in his individual mind provide him with as good a basis for a catalogue raisonné as the notions in the minds of all mankind?

[Mill starts this paragraph with an odd warning against listing and classifying things without using their names. Then:] If we begin with names, and use them as our clue to the things, this brings before us all the distinctions that have been recognised by all inquirers taken together. I think it will be found that mankind have multiplied the varieties unnecessarily, mistaking differences in •the manner of naming things for differences among •things. But we aren’t entitled to assume this at the outset. We must start by recognising the distinctions made by ordinary language. If some of these turn out not to be fundamental, the enumeration of the different kinds of realities can be abridged accordingly; but a logician can’t reasonably start by imposing on the facts the yoke of a theory and reserving the evidence for the theory for discussion later on.

Chapter 2. Names

§1. ‘A name’, says Hobbes, ‘is a word taken at pleasure to serve as a mark that may raise in our mind a thought like some thought we had before, and which being pronounced to others gives them a sign of what thought the speaker has. . . before in his mind.’ This simple definition of a name, as a word or phrase that serves as

•a mark to recall to ourselves the likeness of a former thought, and

•a sign to make it known to others,

seems to be just right. Names do much more than this; but all the rest grows out of the two roles mentioned in Hobbes’s definition. I’ll show this in due course.

In ordinary contexts, names are taken to be names of things; but some metaphysicians have said that they are names of our ideas of things, and have thought this to be a highly important point. Hobbes, for example, writes:

Seeing that names ordered in speech (as is defined) are signs of our conceptions, they are obviously not signs of the things themselves. ‘The sound of the word stone is the sign of a stone’ is true only if it means that the hearer gathers that the speaker is thinking of a stone.

If the point is merely that what is brought back into the speaker’s mind or conveyed to the hearer is not a stone, there’s no denying it. But here is a good reason for sticking by the common usage—as Hobbes himself does in other places—and take the word ‘sun’ to be the name of •the sun and not of •our idea of the sun. Names are intended not only to make the hearer conceive what we •conceive but also to inform him of what we •believe. Now, when I use a name to express a belief, it’s a belief about the thing itself, not about my idea of it. When I say ‘The sun causes daylight’ I don’t mean that my idea of the sun causes in me the idea of daylight; I mean that a certain physical fact (the sun’s presence). . . .causes another physical fact, namely daylight. . . . In this work names will always be spoken of as the names of things and not merely of our ideas of things.

What things? To answer this we must look into the different kinds of names.

§2. It is usual to preface a study of names by distinguishing them from words that aren’t names but only parts of names. These are taken to include

•particles, e.g. ‘of’, ‘to’, ‘truly’, ‘often’;

•the inflected cases of nouns and pronouns, e.g. ‘me’, ‘him’, ‘John’s’; and even

•adjectives, e.g. ‘large’, ‘heavy’.

These words don’t stand for things of which you can affirm or deny anything. We can’t say

‘(A) heavy fell’, ‘A truly was asserted’, ‘(An) of was in the room’.

Unless of course we are speaking about the words themselves:

‘“Truly” is an English word’, ‘“Heavy” is an adjective’.

In that case they are complete names of those particular sounds or series of written characters. Except in that kind of use, these words can only be part of the subject of a proposition, as in ‘A heavy body fell’, ‘A truly important fact was asserted’. . . .

But an adjective can stand by itself as the predicate of a proposition, as in ‘Snow is white’. [Mill then discusses cases where an adjective functions as the subject of a proposition, as in ‘White is an agreeable colour’. This could be done much more freely in Greek and Latin that it can in English, Mill says:] We may say ‘The earth is round’ but not ‘Round is easily moved’ rather than ‘A round object is easily moved’. But this distinction is grammatical rather than logical: ‘round’ has exactly the same meaning as ‘a round object’, and it’s only custom that prescribes which is to be used in a given context. So I shan’t hesitate to call adjectives names. . . . The other classes of subsidiary words have no claim whatever to be regarded as names. An adverb or an accusative case can’t in any context figure as one term in a proposition unless it’s a proposition about that sound or sequence of letters.

[Mill mentions some scholastic technical terms with which he hasn’t much patience. The main content of this paragraph is a sorting out of •words that can be used only as parts of names; •one-word names, i.e. words each of which can, unaided, serve as subject or predicate of a proposition; and •‘many-worded names’, i.e. phrases each of which contains words from each of the other two categories and is itself a name, i.e. can serve as subject or predicate of a proposition. Mill goes on to discuss many-worded names.] A number of words often compose one single name, and no more. A logician will see this phrase:

‘the place which the wisdom or policy of antiquity had destined for the residence of the Abyssinian princes’

as only one name. A test for whether any phrase constitutes one name or more than one is to predicate something of it and then see whether we make only one assertion or several. Consider these:

(a) ‘John Nokes, who was the mayor of the town, died yesterday.’

(b) ‘John Nokes and the mayor of the town died yesterday.’

Of these, (a) makes one assertion, (b) makes two. It’s true that (a) includes another assertion, namely that John Nokes was mayor of the town. But this assertion was already made: we didn’t make it by adding the predicate ‘died yesterday’.

That’s enough about many-worded names. Let us now look into ways of classifying names on the basis not of the words they’re composed of but of their signification.

·FIRST DIVISION: ‘UNIVERSAL’ AND ‘SINGULAR’·

§3. All names are names of something, real or imaginary; but many things haven’t been given individual names. Others have—e.g. persons and remarkable places. And when we want to pick out something that doesn’t have its own individual name, we construct one out of two or more words, each of which could be used by itself to name an indefinite number of other objects. Example: I say ‘this stone’ to designate x, one particular stone; ‘this’ and ‘stone’ are each names that can be applied to many things other than x, though in combination in this particular context they pick out x.

That is one use of names that apply to more than one thing; if it were their only use, they would be mere contrivances for economising the use of language. But it’s obviously not their only function. They also enable us to assert general propositions, affirming or denying a predicate of an indefinite number of things at once. So the distinction between general names and individual or singular names is fundamental, and can be considered as the most basic classificatory split in names. Here’s how it is standardly understood:

•‘general name’ = ‘name that can be truly affirmed, in the same sense, of each of indefinitely many things’.

•‘individual or singular name’ = ‘name that can be truly affirmed, in the same sense, of only one thing’.

Thus, ‘man’ can be truly affirmed of John, George, Mary, and other persons without limit; and it is affirmed of all of them in the same sense; for ‘man’ expresses certain qualities, and when we predicate it of those persons we assert that they all have those qualities. But ‘John’ can be truly affirmed of only one person, at least in the same sense. Many persons have that name, but it isn’t given to them to indicate anything they have in common; it can’t be said to be affirmed of them in any sense, so it’s not affirmed of them in the same sense. ‘The king who succeeded William the Conqueror’ is also an individual name, because the meaning of the words in it imply that there can’t be more than one person of whom it can be truly affirmed. Even ‘the king’, when the occasion or the context picks out the person of whom it is to be understood, can fairly be regarded as an individual name.

Quite often people explain what ‘general name’ means by saying that a general name is the name of a class. This is a convenient thing to say for some purposes, but it won’t do as a definition, because it explains the clearer of two things by the more obscure. It would be more logical to reverse the proposition so that it defines the word ‘class’: ‘A class is the indefinite multitude of individuals denoted by a general name.’

General names must be distinguished from collective names. A general name can be predicated of each individual in a multitude; a collective name can be predicated only of the multitude as a whole. ‘The 76th infantry regiment in the British army’ is a collective name, not general but individual; it can be predicated of a multitude of individual soldiers taken jointly, but not of any individual soldier. . . .

Whereas ‘the 76th regiment’ is a collective name but not a general one, ‘a regiment’ is both collective and general. It’s •general with respect to all individual regiments, of each of which it can be affirmed, and •collective with respect to the individual soldiers in any regiment.

·SECOND DIVISION: ‘CONCRETE’ AND ‘ABSTRACT’·

§4. Our next division of names is into •concrete and •abstract. A concrete name is one that stands for a thing; an abstract name stands for an attribute of a thing. Thus ‘John’, ‘the sea’, ‘this table’, are names of things. ‘White’ is also a name of things, and ‘whiteness’ is the name of a quality or attribute they all have. ‘Man’ is a name of many things; ‘humanity’ is a name of an attribute of those things. ‘Old’ is a name of things; ‘old age’ is a name of one of their attributes.

I have used ‘concrete’ and ‘abstract’ in the sense given them by the scholastics, who—despite the imperfections of their philosophy—were unrivaled in the construction of technical language. They didn’t go very far into logic, yet their definitions in logic have seldom been altered without being spoiled. But more recently a practice has grown up—introduced or at least encouraged by Locke—of applying ‘abstract’ to all names that result from •abstraction or •generalisation, thus counting as ‘abstract’ all general names rather than only the names of attributes. The metaphysicians of the Condillac school have followed Locke in this (they have generally accepted the weakest parts of his philosophy and ignored the best work of that truly original genius), popularising his use of ‘abstract’ to the point where it isn’t easy to restore the word to its original signification. This was a reckless and irresponsible change in the meaning of the word: it leaves us with no compact distinctive name for an important class of words, the names of attributes; and it gives ‘abstract’ a role that was already being performed by the phrase ‘general name’, which has an exact equivalent in every language I am acquainted with. The old meaning, however, hasn’t disappeared so completely that those of us who still adhere to it have no chance of being understood. By ‘abstract’, then, I shall always in logical contexts mean the opposite of ‘concrete’—taking an abstract name to be the name of an attribute, and a concrete name to be the name of an object.

Are abstract names general or singular? Some are general, namely those that are names of a class of attributes. [Mill gives examples, and works his way to the point that he’ll have to count as ‘general’ any word that names an attribute that could be further specified, so that an abstract name is ‘singular’ only if it designates an absolutely utterly specific attribute. He then backs out:] To avoid merely verbal disputes, the best course would probably be to consider these names as neither general nor individual, and to place them in a class apart.

[Mill anticipates the objection that attributes are named not only by the names he has called ‘abstract’ but also by adjectives. He denies this, maintaining that when the noun ‘whiteness’ is used it is to say something about that colour, but that in (for instance) ‘Snow is white’ the topic is not the colour but snow. He concludes:] We’ll soon see that every name that has any signification—any name such that when it is applied to an individual x some information about x is given—implies an attribute of some sort; but it isn’t a name of the attribute, which has its own proper abstract name.

·THIRD DIVISION: ‘CONNOTATIVE’ AND ‘NON-CONNOTATIVE’·

§5. This leads to our next topic, a third great division of names, into those that are and those that are not connotative. . . . This is one of the most important distinctions that I’ll discuss, and is among those that go deepest into the nature of language.

•A term is connotative if it denotes a subject, and implies an attribute.

•It is non-connotative if it merely signifies a subject or an attribute without implying anything about its attributes.

By ‘subject’ here I mean anything that has attributes. Thus ‘John’, ‘London’, and ‘England’ are names that signify a subject only. ‘Whiteness’, ‘length’ and ‘virtue’ signify an attribute only. So none of these names is connotative. But ‘white’, ‘long’ and ‘virtuous’ are connotative. The word ‘white’ denotes all white things—snow, paper, sea-foam etc.—and implies or (in scholastic terminology) connotes[1] the attribute whiteness. The word ‘white’ is not predicated of the attribute but of the things that have it, and we convey that they have it when we predicate ‘white’ of them. This holds also for the other words I have cited. ‘Virtuous’, for example, is the name of a class that includes Socrates, ·prison reformer John· Howard, the ·philanthropist known as the· Man of Ross, and an indefinable number of other past, present and future individuals. ‘Virtuous’ denotes these individuals; it is their name; but it applies to them because of an attribute they are all supposed to have, namely virtue; it is applied to all and only beings that are thought to have this attribute.

All concrete general names are connotative. The word ‘man’ denotes Peter, Jane, John and an indefinite number of other individuals, and is their class-name. Applying it to them is signifying that they have certain attributes— corporeity, animal life, rationality, and what we call (for short) the ‘human’ external form. . . . The word ‘man’ signifies •all these attributes and •all subjects that have them. But it can be predicated only of the subjects. . . . It signifies the subjects directly, the attributes indirectly; it denotes the subjects and implies or involves or indicates or as I shall say from now on connotes the attributes. It is a connotative name.

[Then a paragraph saying that (for example) because snow is given the name ‘white’ because it has the attribute whiteness, the attribute ‘denominates’ snow. That’s the last we hear of that unpromising thought.]

All concrete general names, then, are connotative. Even abstract names, though they name only attributes, may in some cases also be connotative. That’s because attributes may have attributes, and a word that •denotes an attribute may •connote an attribute of it. Consider the word ‘fault’ = ‘bad or hurtful quality’. This word is a name common to many attributes, and connotes hurtfulness, an attribute of them. When we say that slowness in a horse ‘is a fault’, we don’t mean that the slow movement, the actual change of place of the slow horse, is a bad thing, but that the property or peculiarity of the horse, from which it derives that name, the quality of being a slow mover, is an undesirable peculiarity. [The last sentence of that is verbatim from Mill.]

In regard to concrete names that aren’t general but individual, a distinction must be made.

Proper names are not connotative: they don’t indicate or imply anything about the attributes of the individuals who bear them. When we name a child ‘Paul’ or a dog ‘Caesar’, these names are simply marks enabling us to say things about those individuals. We presumably had some reason for our selection of a name for a given individual, but the name it has been given it is independent of the reason. . . . A town may have been named ‘Dartmouth’ because it is at the mouth of the river Dart, but its name doesn’t mean that. If an earthquake changed the river’s course, putting a distance between it and the town, the town’s name would not have to be changed. . . . A proper name is attached to the object itself, and doesn’t depend on the continuance of any attribute of the object.

An individual name—i.e. one applicable to only one object—may be really connotative. We can give an individual a utterly unmeaning name, a proper name that serves to pick a thing out without saying anything about it, but a name applying to just one individual isn’t necessarily like that. [Mill mentions ‘the sun’, and ‘God’ in the mouth of a monotheist, and says they may look like examples of what he is talking about but really aren’t. That’s because it’s not a fact about either of those names that it applies to only one individual. We can imagine a world where there are many suns, and some people believe that there actually are many gods. He then moves ahead:] It is easy to produce real instances of connotative individual names. The very meaning of the connotative name may imply that there can’t be two individuals with the attribute it connotes: e.g. ‘the only son of John Stiles’, ‘the first emperor of Rome’. Or the attribute connoted may be a connection with a particular event, and it may be that only one individual could have that relation to that event (‘the father of Socrates’) or that only one individual actually did have it (‘the author of the Iliad’, ‘the murderer of Henri IV’). What is here done by the word ‘the’ is done in other cases by the context: ‘Caesar’s army’ is an individual name if the context shows that what is meant is the army that Caesar commanded in a particular battle. . . . And I have already mentioned another quite common case: A many-worded name can consist of •a general name that can be applied to more than one thing plus •other words that limit the general name so that the entire expression can be applied to only one object. An example: ‘Prime Minister of England’ is a general name; but at any given time ‘present Prime Minister of England’ can pick out only one person. Taking it that it’s a fact about the meaning of ‘Prime Minister of England’ that it can’t apply to two people at once, we can say that the singularity of ‘present Prime Minister of England’ is secured by •its meaning, without bringing in •any extrinsic facts; so it is strictly an individual name.

It’s easy to see from all this that whenever the names given to objects convey any information—i.e. whenever they have properly any meaning—the meaning resides not in what they denote but in what they connote. The only names of objects that connote nothing are proper names; and strictly speaking these have no signification.

. . . .When we give something a proper name, what we are doing is like what a robber does when he puts a chalk-mark on a house so that he will recognise it when he next comes into this street. We put a mark not actually on the object itself, but on the idea of the object, so to speak. A proper name is just an unmeaning mark that we connect with the idea of the object, so that when the mark meets our eyes or occurs to our thoughts we will think of that object. . . .

When we apply a proper name to a thing—‘That’s Smith over there’, ‘This is York’—we aren’t giving the hearer any information about Smith or York except that those are their names. You may come to have more information, e.g. because you knew already that York contains the Minster. But your new knowledge that this town here contains the Minster doesn’t come from anything implied in the name. With connotative names it’s a different situation. ‘This town is built of marble’—that may give the hearer new information that comes from the signification of the many-worded connotative name ‘built of marble’. Such names. . . .aren’t mere marks; they are significant marks, and the connotation is what constitutes their significance.

A connotative name should be thought of as a name of all the various individuals that it is predicable of, i.e. that it denotes, and not of what it connotes. (I have given reasons for this; another reason is that it preserves a certain analogy between connotative and proper names.) But by learning •what things it names we don’t learn •the meaning of the name. [Mill discusses the different descriptions— ‘connotative names’—that we could give to Sophroniscus, the father of Socrates. His main point is to highlight the difference between •knowing that a certain expression applies to Sophroniscus and •knowing what the expression means. He sums up:] It could even happen that I know every single individual to whom a given ·connotative· name applied and yet not know the name’s meaning. . . .

It’s sometimes hard to decide precisely how much a particular word connotes: we don’t know exactly—because we haven’t needed to decide—how much difference in the object would require a different name for it. Obviously ‘man’ connotes along with animal life and rationality, a certain external form; but exactly what form? How different from us, physically, would a newly discovered race have to be for it not to count as ‘human’? Again, rationality is a quality that admits of degrees, and it has never been settled what is the lowest degree of rationality that would entitle any creature to count as a human being. In any case like this the meaning of a general name is unsettled and vague because mankind haven’t come to any positive agreement about it, I’ll show later (when discussing classification) the conditions in which this vagueness is not inconvenient; and I’ll present cases where vagueness serves language’s purposes better than complete precision. . . .

But this partial uncertainty about the connotation of names is troublesome unless guarded against by strict precautions. Lax habits of thought are largely due to the custom of using connotative terms. . . .with no more precise notion of their meaning than can be loosely gathered from seeing what objects they are used to denote. This is how, inevitably, we get our first knowledge of our first language: a child learns the meanings of ‘man’ and ‘white’ by •hearing them applied to a variety of objects, and •discovering (by a process of generalisation and analysis that he can carry out but couldn’t describe) what those objects have in common. In many cases, including those two words, the process is so easy that it doesn’t need assistance from culture [Mill’s phrase]. . . . But in many other cases, objects are classified together in common speech because of their general resemblance to one another, but it’s not easy to say exactly what the attributes are that create this resemblance; working that out requires more analytic habits than most people possess. When this is the case, people use the name without any precise meaning; they talk—and therefore think—vaguely, giving about as much significance as a three-year-old attaches to ‘brother’ and ‘sister’. [The child, Mill says, isn’t in trouble here because in cases of doubt there’s usually an authority to tell him whether a new individual is a brother or sister. Not so, however, in most cases of meaning-doubt, when each of us has to decide for him- or herself how a new item is to be classified. He continues:] So we do this on the basis of superficial similarity between the new object and familiar objects already named. For example, an unknown substance found in the ground will be called ‘earth’, ‘sand’, or ‘a stone’, according to its texture. And so names creep on from subject to subject until (sometimes) all traces of a common meaning disappear and the word comes to denote a number of things. . . .that have no attribute in common—or only an attribute shared by other things to which the name is arbitrarily refused. [A footnote here quotes from Bain’s Logic a lively passage about the irrationality of the facts about how the word ‘stone’ is/isn’t applied.] Even scientific writers have joined in this pushing of general language away from its purpose; sometimes because. . . .they knew no better, and sometimes because of a general reluctance to admit new words. This attitude leads mankind in non-technical subjects to try to make the original stock of names serve with little augmentation to express an increasing number of objects and distinctions, and thus to make an increasingly bad job of expressing them.

Anyone who has thought hard about the present condition of mental and moral philosophy knows how far this loose way of classifying and naming objects has impeded accurate thinking in those areas. Should we then introduce a new technical language to use in theorising in them? No: that would be extremely difficult to do, and would have considerable drawbacks if it were done, because the topics of mental and moral philosophy are also topics of everyday informal conversation. So the philosopher faces the problem—one of the hardest he has to solve—of retaining the existing phraseology while lessening its imperfections. The only way to do this is to give to every often-used general concrete name a definite and fixed connotation, so that when we call an object by that name it will be known what attributes we mean to predicate of it. This will be a delicate operation if the newly fixed connotation is to

•make the least possible change in the objects the name is habitually employed to denote—the least possible addition to or subtraction from the group of objects which it has, perhaps imperfectly, served to mark off and hold together; and to

•do the least damage to the truth of any propositions that are commonly accepted as true.

That’s what people are aiming at when they try to define a general name that is already in use. . . . No questions in the moral sciences have aroused keener controversy than the definitions of almost all the leading expressions, which shows what a large problem this is.

Names with •indeterminate connotation musn’t be confused with names that have •more than one connotation, i.e. with ambiguous words. A word may have several meanings, but all of them fixed and recognized—for example ‘post’ and ‘box’, the various senses of which it would take for ever to enumerate. And because demand for names often outruns supply, it is sometimes advisable—even necessary— to retain a name with all these meanings, distinguishing the meanings clearly enough to prevent their being confused with one another. We can regard such a word as two or more names that happen to be written and spoken alike. [At this point Mill has a long footnote disagreeing with James Mill, his father, an authority that ‘I am less likely than any other person to undervalue’, about the best way to use ‘connotation’. The subsequent history of the word followed the son rather than the father, and the details of the disagreement are not now interesting.]

·FOURTH DIVISION: ‘POSITIVE’ AND ‘NEGATIVE’·

§6. The fourth principal division of names is into •positive and •negative. Positive like ‘man’, ‘tree’, ‘good’; negative like ‘not-man’, ‘not-tree’, ‘not-good’. To every positive concrete name Np we could have a second name Nn to apply to all and only the things that Np doesn’t apply to. . . . When the positive name is connotative, the corresponding negative name is also connotative, but in a special way, connoting not the presence but the absence of an attribute. Thus, ‘not-white’ connotes the attribute of not possessing whiteness—yes, that is an attribute too.. . . .[2]

Many names that are positive in form are negative in reality, and others are negative in form but really positive. The word ‘inconvenient’, for example, doesn’t express the mere absence of convenience; it expresses a positive attribute— that of being the cause of discomfort or annoyance. So the word ‘unpleasant’, despite its negative form, doesn’t connote the mere absence of pleasantness, but a lesser degree of what is ‘painful’, which is obviously positive. And ‘idle’ is positive in form, expressing nothing but what would be signified either by ‘not working’ or ‘not disposed to work’; and ‘sober’ by ‘not drunk’. . . . [Mill seems to have committed himself to allowing that the moon is convenient, the number 99 is pleasant, beach sand is idle, and the Nile is sober. These positive/negative oppositions seem to fit Bain’s account (preceding footnote) rather than Mill’s.]

There’s a class of names called ‘privative’. A privative name has the same signification as a •positive and a •negative name taken together: it’s the name of something that used to have (or might have been expected to have) an attribute that it actually doesn’t now have. An example is ‘blind’: it doesn’t mean merely the negative ‘doesn’t see’ or ‘can’t see’, because sticks and stones aren’t literally ‘blind’. Something isn’t usually said to be ‘blind’ unless (a) the class of things it is related to, either usually or on this particular occasion, is chiefly composed of things that can see—e.g. ‘blind man’ or ‘blind horse’; or (b) it is supposed for some reason that it ought to see—e.g. saying that a man ‘rushed blindly into an abyss’, or that most philosophers or clergy ‘are blind guides’. So ‘privative’ names connote •the absence of certain attributes and •the presence of others from which the presence also of the former might naturally have been expected.

·FIFTH DIVISION: ‘RELATIVE’ AND ‘NON-RELATIVE’·

§7. The fifth leading division of names is into •relative and •non-relative. Names of the latter kind are sometimes called ‘absolute’; but this word is so hard at work in metaphysics that we should spare it when we can do without it. It resembles the word ‘civil’ in the language of jurisprudence, which stands for the opposite of •‘criminal’, of •‘ecclesiastical’, •of ‘military’, •of ‘political’—in short, the opposite of any positive word that lacks a negative.

Here are some relative names:

‘father’, ‘son’;

‘ruler’, ‘subject’;

‘like’, ‘unlike’;

‘equal’, ‘unequal;

‘longer’, ‘shorter’

‘cause’, ‘effect’.

They are always given in pairs: Every relative name Nr that is predicated of an object presupposes another object (or objects) of which we may predicate either Nr or some relative name that we take to be the correlative of Nr. Calling someone a ‘son’, we suppose other persons who are his parents. Calling an event a ‘cause’, we suppose another event that is an effect. . . . When we call someone ‘a sibling’ we suppose someone else who is also a sibling. In this last case the relative term is its own correlative. [The ‘sibling’ example replaces one of Mill’s which is no longer correct English.]

Concrete names of this kind are, like other concrete general names, connotative; they •denote a subject and •connote an attribute; and each of them has or could have a corresponding abstract name to denote the attribute connoted by the concrete. Thus the concrete ‘sibling’ has its abstract ‘siblinghood’; ‘father’ and ‘son’ do or could have the abstracts ‘paternity’ and ‘sonship’. The concrete name connotes an attribute, and the corresponding abstract name denotes that attribute. But what sort of attribute? What’s the special feature of the connotation of a relative name?

Some say that the attribute signified by a name is a relation. Even if this doesn’t explain much, they think, it is the best answer we can get. If we ask ‘Well, then, what is a relation?’ they don’t profess to have an answer. It is generally thought that a relation is something particularly specialised and mysterious; but I can’t see what makes it more so than any other attribute—it seems to me indeed to be somewhat less obscure than other attributes. In my opinion, the best way to get a clear insight into the nature of all attributes (i.e. of all that is meant by an attribute) is through an examination of the signification of relative names (i.e. of the nature of the attribute that they connote). [Mill’s odd phrase ‘all that is meant by an attribute’ probably means ‘all that is meant by “attribute”’.]

Take the correlative names ‘father’ and ‘son’: they denote different objects, but in a certain sense they connote the same thing. They don’t connote the same attribute, but the two propositions

•A is the father of B and

•B is a son of A

say exactly the same thing, express the very same fact. When that fact is analysed, we find that it consists of a series of physical events. . . .in which both A and B are involved and from which they both derive names. What those names really connote is this series of events: that is the whole meaning that each of them is intended to convey. The series of events can be said to constitute the relation. . . . It seems that all we need to account for the existence of relative names is that whenever there’s a fact in which two individuals are involved, each individual has an attribute grounded on that fact.

Here are three equivalent accounts of what it means to say that a name N of something x is ‘relative’:

•in addition to x, N implies in its signification the existence of another object which also gets a name from the fact that is the ground of N;

•N is the name of x but its signification can’t be explained without mentioning something else;

•N can’t be meaningfully employed in discourse unless the name of something other than x is also expressed or presupposed.

These definitions are all basically equivalent, being different ways of expressing this one distinctive circumstance: if we think of a state of affairs in which everything except x goes out of existence,[3] we can consistently suppose that x still keeps all its non-relative attributes but we can’t consistently suppose that it keeps any of its relative attributes.

§8. Names have also been further distinguished into unambiguous and ambiguous; these, though, are not two kinds of •names but two •ways of using names. A name is applied unambiguously with respect to all the things it can be applied to in the same sense; it is applied ambiguously with respect to the things that it is applied to in different senses. You hardly need examples of a phenomenon as familiar as a word with a double meaning. In reality, as I remarked here, an ambiguous word is not one name but two that happen to sound the same: ‘file’ meaning a steel instrument and ‘file’ meaning a line of soldiers have no more right to be regarded as one word because they are written alike than ‘grease’ and ‘Greece’ have because they sound alike. . . .

[Mill mentions the case where a word’s different meanings have something in common, and where one of the two strikes us as primary, and the other as secondary or metaphorical; for example the adjective in ‘brilliant light’ and ‘brilliant achievement’. He continues:] In these cases, however, the two-names-with-the-same-sound diagnosis applies just as well as in cases of perfect ambiguity. A very common form of fallacious reasoning arising from ambiguity is that of arguing from a metaphorical expression as if it were literal. . . .

NOTES

[1] Notare [Latin], to mark; connotare, to mark along with; to mark one thing with or in addition to another.

[2] Bain in his Logic says that negative names are names only of some particular class of things that aren’t denoted by the corresponding positive name. He holds, for instance, that ‘not-white’ applies only to every coloured thing that isn’t white. But here, as everywhere, the test of •what a name denotes is •what it can be predicated of; and we can certainly say of a sound or a smell that it is not white. The affirmation and the negation of the same attribute have to divide the whole field of predication between them.

[3] Or rather, everything except x and the percipient mind. I’ll show later that to ascribe any attribute to an object necessarily implies a mind to perceive it.—The simple and clear explanation I have given relations and relative names, a topic that has for so long been vexatious in metaphysics, was first given, I think, by James Mill in his Analysis of the Phenomena of the Human Mind.

Chapter 3. Things that are denoted by names

§1. Let’s retrace our steps up to here. Logic is the theory of proof, which presupposes something provable, which must be a proposition or assertion. . . ., which is discourse that affirms or denies something of something else. These two things are signified by two names which when joined by a copula constitute the proposition. I reviewed in chapter 2 the various kinds of names so as to ascertain what each signifies, taking this far enough to be able to. . . .enumerate all the kinds of things that can be predicates or of subjects of predication. With that done, it can’t be very hard to determine the import of predication, i.e. of propositions.

The scholastics were aware of the need to enumerate existences as the basis of logic, and so was their master Aristotle, the most comprehensive of the ancient philosophers if not also the wisest. The Categories. . . .were believed to be an enumeration of all the things that could be named; an enumeration by the summa genera [see Glossary], i.e. the most extensive classes into which things could be distributed; which, therefore, were so many highest Predicates, one or other of which they thought could be truly affirmed of every nameable thing whatsoever. Here are the classes into which (according to this school of philosophy) things in general might be reduced [Mill gives them in Greek and Latin]:

substance

quantity

quality

relation

action

passivity

position in space

position in time

posture

state

The imperfections of this system are too obvious to need a detailed examination, and it isn’t good enough to make such an examination worthwhile. It’s a mere list of the distinctions roughly marked out by the language of everyday life, with little philosophical analysis to reveal their rationale. Even a superficial analysis would have shown the list to be both redundant (it contains some things more than once) and defective (it omits some objects that should be there). ·Redundancy:· It’s like a division of animals into men, quadrupeds, horses, asses, and ponies! It puts action and passivity alongside relation. . . . [Mill makes similar remarks about some of the others; but they aren’t very clear, and we can do without them.] ·Omissions:· The list ignores everything but substances and attributes. Which category is supposed to contain sensations, or any other feelings and states of mind—hope, joy, fear; sound, smell, taste; pain, pleasure; thought, judgment, conception, and the like?. . . . Feelings or states of consciousness are assuredly to be counted among realities, but they can’t be reckoned among either substances or attributes.

[Mill has here a long footnote discussing a comment by Bain on the preceding passage. Bain says that Aristotle didn’t intend the Categories as kinds of things but as kinds of questions, so that Mill’s criticisms miss the mark. Mill accepts this, but adds:] So Aristotle may not have seen the Categories as classification of things; but his scholastic followers certainly did see them that way, and employed them on that basis: subdividing •the category substance, as a naturalist might do, into the different classes of physical or metaphysical objects as distinguished from attributes, and subdividing •the other categories into the principal varieties of quantity, quality, relation, etc. So it is fair to complain that they had no category of feeling. Feeling is assuredly predicable, as a summum genus, of every particular kind of feeling (e.g. Bain’s example of hope): but it can’t be brought within any of the Categories as interpreted by Aristotle or by his followers.

§2. Before starting to re-do this work, making a better job of it than the early logicians did, I must comment on an unfortunate ambiguity in all the concrete names that correspond to ‘existence’, the most general of all abstract terms. When we need a name to denote anything that exists—as distinct from non-entity or Nothing—almost every word that could do this also has an even more familiar sense in which it denotes only substances. But substances aren’t all that exists. If attributes are to be spoken of, they must be said to exist; feelings certainly exist. Yet when we speak of an ‘object’ or of a ‘thing’ we are almost always supposed to mean a substance. There seems to be a kind of contradiction in saying that one thing is merely an attribute of another thing. . . . If we reject ‘thing’ and look for a word whose dominant or only meaning makes it •denote whatever exists and •connote only simple existence, it might seem that the best word for this purpose is ‘being’. It is basically the present participle of the verb ‘to be’, which in one of its meanings is exactly equivalent to the verb ‘to exist’; so that even the grammatical formation of ‘being’ makes it suitable to be the concrete ·term· corresponding to the abstract ·term· ‘existence’. But oddly enough this word is even more completely spoiled than ‘thing’ for the purpose that it seemed perfectly made for. ‘Being’ is customarily exactly synonymous with ‘substance’ (except that it can be applied equally to matter and to mind, whereas ‘substance’, though originally and strictly applicable to both, is apt to suggest the idea of matter). Attributes are never called ‘beings’; nor are feelings. A ‘being’ is something that arouses feelings and has attributes. The soul is called a ‘being’; God and angels are called ‘beings’; but if we said that extension, colour, wisdom and virtue are ‘beings’ we might be suspected of thinking

•with some of the ancients, that the cardinal [see Glossary] virtues are animals; or

•with the Platonic school that ‘Ideas’—·which play the role of attributes·— can exist without anything having or instantiating them; or

•with the followers of Epicurus that sensible forms— ·which they thought to be instances of attributes·— radiate out in every direction from bodies and by coming in contact with our organs cause our perceptions;

i.e. of thinking that attributes are substances!

Philosophers looking for something to replace the spoiled ‘being’ laid their hands on the word ‘entity’. This piece of barbarous Latin was invented by the scholastics as an abstract name, which is what its grammatical form seems to make it; but being seized by logicians in distress to stop a leak in their terminology, it has ever since been used as a concrete name. [To make sure that that is understood: The scholastics used ‘entity’ to mean ‘existence’, i.e. what existing items have; the panicking logicians used it mean ‘existent’, i.e. what existing things are. Incidentally, Mill writes as though he weren’t aware that ‘existent’ was available as a noun to do exactly the work he wanted done.] (The related word ‘essence’, born at the same time and of the same parents, scarcely underwent a more complete transformation when it went from being the •abstract noun of the verb ‘to be’ to denoting something •concrete enough to be kept in a glass bottle! [This is a joking allusion to ‘vanilla essence’ etc. on grocery shelves.]) After the word ‘entity’ settled down into a concrete name, it kept its universality of signification rather better than any of the other names I have mentioned. But all the language of psychology seems liable to gradual decay as it gets older, and ‘entity’ hasn’t been exempt from that. If you call virtue an ‘entity’ you aren’t as strongly suspected of believing it to be a substance as you would be if you called it a ‘being’; but you’re still somewhat open to that suspicion. Every word that was originally intended to connote mere •existence seems in the course of time to enlarge its connotation to •separate existence, i.e. existence freed from the condition of belonging to a substance; which has the effect of shutting out attributes, and along with them feelings, which in nearly every case have no name except that of the attribute grounded on them. The greatest difficulty confronting those who have a lot of thoughts to express is finding a sufficient variety of precise words fitted to express them; which makes it strange that people—even scientific thinkers—are seriously addicted to the practice of taking valuable words to express ideas that are already well enough expressed by other words!

When we can’t get good tools, the next best thing is to understand the defects of the tools we have. That’s why I have warned you of the ambiguity of the names that I have to employ for want of anything better. I must now try to use them in a way that won’t ever leave my meaning doubtful or obscure. Because none of the above words is altogether unambiguous, I shan’t restrict myself to any one; rather, on each occasion I’ll use the word that seems least likely in that particular case to lead to misunderstanding. I don’t claim to keep these or any other words strictly to one single sense. Doing that would often leave me without a word to express something that is signified by a known word in one or other of its senses. . . . It wouldn’t be wise for me, when writing on such an abstract subject, to deny myself the advantage I can get from even an improper use of a term, when by using it I can call up some familiar association that will bring my meaning home to your mind in a flash.

The difficulty that you and I will both have in getting vague words to convey a precise meaning is not wholly a matter for regret. . . . Philosophical language will for a long time, and everyday language still longer, retain so much vagueness and ambiguity that logic wouldn’t be worth much if it didn’t, among its other advantages, exercise the understanding in doing its work neatly and correctly with these imperfect tools.

Now it is time to proceed to my list. I shall start with feelings, the simplest class of nameable things—the word ‘feeling’ being of course understood in its broadest sense.

I. FEELINGS, OR STATES OF CONSCIOUSNESS.

§3. In the language of philosophy, ‘feeling’ and ‘state of consciousness’ are equivalent expressions: everything the mind is conscious of is a feeling. . . . In everyday language ‘feeling’ is not always synonymous with ‘state of consciousness’: it is often taken more restrictedly for states that are thought of as belonging to the sensitive or the emotional side of our nature, and sometimes even more restrictedly to the emotional side alone, excluding anything thought of as belonging to the percipient or to the intellectual side. But this is an admitted departure from correctness of language. [Mill mentions comparable ‘perversions’ of language: using ‘mind’ to refer only to the intellect; using ‘feeling’ to refer only to tactual sensations.]

In the proper sense of the term, feeling is a genus of which sensation, emotion and thought are species. I am taking ‘thought’ to cover everything we are internally conscious of when we are said to think: from •the consciousness we have when we think of a red colour without having it before our eyes to •the most difficult thoughts of a philosopher or poet. Bear in mind that a thought is what occurs in the mind itself; what a person is thinking about isn’t a thought. . . . Even imaginary objects. . . .are distinct from our ideas of them. I may think of a hobgoblin, or the loaf I ate yesterday, or the flower that will bloom to-morrow. These things •never existed or •no longer exist or •don’t yet exist; they are distinct from my thought which exists right now.

Similarly, a sensation should be carefully distinguished from the object that causes it—our •sensation of white is distinct from •a white object. And equally distinct from •the attribute whiteness, which we ascribe to the object because it causes the sensation. It’s hard for us to be clear and discriminating in dealing with these subjects because our sensations seldom have separate names of their own. We have

•a name for the objects that produce a certain sensation in us: ‘white’.

We also have

•a name for the quality in those objects that causes the sensation: ‘whiteness’.

But when we want

•a name for the sensation itself we have to say ‘the sensation of white’ or ‘. . . of whiteness’.

We have to name the sensation from the causing object or the causing attribute. If the sensation arose spontaneously in the mind without anything causing it, which it conceivably could, we would be at a loss. (This isn’t surprising. The need to talk about the sensation arises only in scientific theorising, and language mostly adapts itself to the common uses of life, which is why it hasn’t given us any single-worded or immediate name for the sensation.) For our sensations of hearing we are better provided: we have the word ‘sound’ and a whole vocabulary of words denoting the various kinds of sounds. Because we are often conscious of these sensations in the absence of any perceptible object, we can more easily conceive having them in the absence of any object whatever. If we shut our eyes and listen to music, we can conceive of a universe with nothing in it except •sounds and •ourselves hearing them; and what is easily conceived separately easily gets a separate name. But in general our names of sensations denote indiscriminately the sensation and the attribute. . . .

[The admirable Mill has just made a common mistake. Perhaps ‘sound’ can refer to auditory sensations, but in its dominant meaning it refers to items out there in the world, with locations and sizes and (at least in theory) shapes. ‘Did you hear that sound a few minutes ago?’ ‘I wonder if that sound was heard on the other side of the island.’ And so on. Mill’s own usage shows this, though he doesn’t notice what he’s doing: ‘. . . a universe with nothing in it except •sounds and •ourselves hearing them’—he presumably doesn’t mean ‘a universe with nothing in it except auditory sensations and ourselves hearing them’! Incidentally, the old question ‘If a tree fell in the forest when there was no-one to hear it, would it make a sound?’ is often presented as deep and puzzling; but it’s actually childish nonsense that feeds on the ambiguity of ‘sound’. If the word is used to refer to sensations, then the answer to the ‘puzzle’ is boringly No. If it refers to objective sounds out there in the world, the answer is boringly Yes.]

§4. A lot of intellectual damage is often done by people’s confusing •the sensation itself with •the bodily state that produces the sensation. One source of confusion in this area is the common division of feelings into •bodily and •mental. There’s no basis for this distinction: even sensations are states of the sentient mind, not states of the body. What I am conscious of when I see the colour blue is a feeling of blue colour, which is one thing; the picture on my retina—or the relevant sequence of events in my optic nerve or in my brain—is another thing, of which I’m not conscious and wouldn’t have known about if it weren’t for scientific investigation. These are states of my body; but the resultant sensation of blue is not a state of body but of the thing that perceives and is conscious, namely mind. When sensations are called ‘bodily feelings’, that’s because they are immediately triggered by bodily states; whereas the other kinds of feelings—e.g. thoughts and emotions—are immediately aroused not by anything acting on the bodily organs but by sensations or by previous thoughts. But this is a distinction in the agency that produces our feelings, not in the feelings themselves, which are all states of mind.

[Mill now gives two paragraphs to the view of some philosophers that in addition to •the action of the outside world on our bodily organs and •the action of the organs in producing a sensation there is •‘a third link in the chain of phenomena, which they call a perception’. He declines to theorise in detail about these perceptions, because he is sure of just one thing about them, and that is enough to make them no business of logic. Thus:] In these so-called perceptions or direct recognitions of objects. . . .external to the mind, I can see only cases of belief that claims to be intuitive, or independent of external evidence. When a stone lies before me, I am conscious of certain sensations that I receive from it; but if I say that

•these sensations come to me from an external object which I perceive,

the meaning of these words is that

•in receiving the sensations I intuitively believe that an external cause of those sensations exists.

As I have said several times—·e.g. here·—the laws of intuitive belief and the conditions under which it is legitimate are a subject that belongs not to logic but to the science of the ultimate laws of the human mind.

The same is true of everything that can be said about the distinction the German metaphysicians and their French and English followers so elaborately draw between the •acts of the mind and its merely •passive states; between what it •gives to and what it •receives from the crude materials of its experience. I’m aware that in the context of those writers’ view of the primary elements of thought and knowledge this distinction is fundamental. My present purpose, however, is to examine not the basic groundwork of our knowledge but how we come by the part of it that isn’t basic; so here the difference between active and passive states of mind is of secondary importance. They are all states of mind, all feelings, by which (I repeat) I mean nothing about their being passive; all I mean is that they are psychological facts, that take place in the mind, and should be carefully distinguished from the physical facts that may be connected with them as effects or as causes.

§5. One kind of active state of mind does merit particular attention because it’s a principal part of the connotation of some important classes of names. I mean •volitions, i.e. •acts of the will. In many cases, when we apply a relative name to a sentient being, much of the name’s connotation consists of that being’s actions—past, present, and possible or probable future. Consider ‘sovereign’ and ‘subject’. The meaning these words convey is of innumerable actions done or to be done by the sovereign and the subject in relation to one another. Similarly with ‘physician’ and ‘patient’, ‘leader’ and ‘follower’, ‘tutor’ and ‘pupil’. Many relative words also connote actions that would be done under certain conditions by other persons: ‘mortgagor’ and ‘mortgagee’, ‘obligor’ and ‘obligee’, and many other words expressing some legal relation, which connote what a court of justice would do if. . . etc. And some relative words connote actions previously done by other persons: e.g. ‘brother’. These examples may show you how much of the connotation of names consists of actions. Now what is an action? Not one thing, but a series of two things: the state of mind called a volition, followed by an effect. The volition or intention to produce the effect is one thing; the effect produced by the intention is another thing; the two together constitute the action. . . .

§6. Of the first leading division of nameable things, namely feelings or states of consciousness, I began by recognising three subdivisions—sensations, thoughts, and emotions. I have illustrated the first two of these at some length; the third, emotions, doesn’t require examples in the same way because it isn’t tangled by ambiguities. And, finally, I needed to add to these three a fourth species, commonly called ‘volitions’. I’ll now move on to the two remaining classes of nameable things: all things that are regarded as external to the mind are thought of as either •substances or •attributes.

II. SUBSTANCES.

Logicians have tried to define ‘substance’ and ‘attribute’, but their definitions are not so much attempts to say what substances and attributes are as instructions about how the grammatical structure of a sentence depends on whether we are speaking of substances or of attributes. Such definitions are language-lessons rather than lessons in mental philosophy. An attribute, say the scholastic logicians, (a) must be the attribute of something: colour must be the colour of something, goodness must be the goodness of something; and (b) if this ‘something’ should cease to exist or cease to be connected with the attribute, the attribute would go out of existence. In contrast with that, a substance is self-existent; we can speak about it without putting ‘of’ after its name. A stone is not the stone of anything; the moon is not the moon of anything—it’s simply the moon. Unless the name we give the substance a relative name, in which case it must be followed by ‘of’ or by some other particle implying a reference to something else; but the substance would still fail test (b) for being an attribute, because it could stay in existence if the ‘something else’ were destroyed. A father must be the father of something; and we might say ‘If there were no child there would be no father’, but this means merely that if there were no child we wouldn’t call this person a ‘father’. The man could still exist even if there were no child; there would be no contradiction in supposing him to exist though everything else in the universe was destroyed. But if you destroy all white substances, where would the attribute whiteness be? Whiteness without any white thing is a contradiction in terms.

That’s the nearest that ordinary logic textbooks come to solving this difficulty. You’ll hardly think it is a satisfactory solution. (a) If an attribute is distinguished from a substance by being the attribute of something, we need to be told what ‘of’ means: it stands in such great need of explanation that it can’t be placed in front of the explanation of anything else. (b) It’s true that a substance can be conceived to exist without any other substance, but it’s also true that an attribute can exist without any other attribute; and although we can’t imagine attributes without a substance, we also can’t imagine a substance without attributes.

Metaphysicians, however, have gone deeper and given a much better account of substance than that. Substances are usually divided into •bodies and •minds, and philosophers have. . . .provided a seemingly flawless definition of each of these.

§7. According to the accepted doctrine of modern metaphysicians, body can be defined as the external cause to which we ascribe our sensations. When I see and touch a piece of gold, I’m aware of a sensation of yellow colour, and sensations of hardness and weight; and by manipulating the gold I can add to these sensations many others. . . . The sensations are all that I’m directly conscious of, but I consider them as produced by something that is independent of my will and external to my body and my mind. This external something I call a ‘body’.

You may want to ask: How do we come to ascribe our sensations to any external cause? And have we sufficient ground for this? Some metaphysicians have started a controversy about this, maintaining that we aren’t justified in referring our sensations to what we call ‘body’ or to any other external cause. I’m not concerned here with this controversy or with the metaphysical intricacies on which it depends; but one of the best ways of showing what ‘substance’ means is to consider how one would have to go about maintaining the existence of body against opponents.

This much is certain: our notion of a body includes the notion of a number of sensations of ours or of other sentient beings, habitually occurring simultaneously. My conception of the table at which I am writing is compounded of

•its visible form and size, which are complex sensations of sight;

•its tangible form and size, which are complex sensations of our organs of touch and of our muscles;

•its weight, which is also a sensation of touch and of the muscles;

•its colour, which is a sensation of sight;

•its hardness, which is a sensation of the muscles;

•its composition, a word that stands for all the varieties of sensation that we receive under various circumstances from the wood of which it is made,

and so on. All or most of these sensations are •often experienced simultaneously, and we learn by experience that they could •always be so;. . . .so that the thought of any one of them makes us think of the others, and the whole becomes mentally amalgamated into one mixed state of consciousness which the followers of Locke and Hartley call a ‘complex idea’.

[From here to † on the next page the discussion is riddled with a mistake nearly all metaphysicians made through several centuries, though it is now generally recognised to be a mistake. It consists in failing to distinguish these two theses:

(1) In addition to all the attributes of a thing there is an underlying sheer pure thing or substratum that has them.

(2) In addition to all our ideas or sensations relating to a thing there is a strictly external non-mental thing that causes them.

Mill switches back and forth between these; the switches won’t be tagged, but the present note should enable you to spot them. For a full discussion see www.earlymoderntexts.com/jfb/substrat.pdf. Here Mill briefly confronts the distinction between •properties and •sensations, and declares it to be merely verbal. Coming where it does, the declaration is astonishing; but Mill moves straight on to argue that it’s just a mistake to think that as well as sensations there are also qualities. This passage involves a different mistake from the one mentioned above— a once-common mistake concerning so-called ‘secondary qualities’. It would take too much space to explain it here, but you might find www.earlymoderntexts.com/jfb/lf6p.pdf helpful. And when you read what Mill says about this here ask yourself this: if he had illustrated his thesis about ‘qualities’ in general with squareness instead of whiteness, how much of its plausibility would remain?]

Some philosophers—·most notably Berkeley·—have argued as follows:

If we conceive an orange to lose its natural colour without acquiring any new one; to lose its softness without becoming hard, its roundness without acquiring any other shape; to lose its size, weight, taste, smell; to lose all its mechanical and chemical properties without acquiring new ones; to become— in short—invisible, intangible, imperceptible by our senses or the senses of any other real or possible sentient beings; nothing would remain. If anything were left, what could its nature be? and how could it manifest its presence? By the evidence of the senses? No, because nothing is apparent to the senses except the sensations. We know that these sensations are bound together by some law; they don’t come together at random, but according to a systematic order that is part of the order established in the universe. When we experience one of these sensations, we usually experience the others also, or know that we have it in our power to experience them. But a fixed law of connection that makes the sensations occur together doesn’t necessarily require what is called a ‘substratum’ to support them. The conception of a substratum is merely one of many possible forms in which that connection presents itself to our imagination. . . . If there were such a substratum, suppose it to be at this instant miraculously annihilated, while the sensations continue to occur in the same order—how would the substratum be missed? What evidence could we have that it had gone out of existence? Wouldn’t we have as much reason to believe that it still existed as we now have? And if we wouldn’t be justified in believing it •then, how can we be so •now? So a body is not anything intrinsically different from the sensations that the body is said to produce in us; it is, in short, a set of sensations—or rather of possibilities of sensation—joined together according to a fixed law.

The controversies these speculations gave rise to, and the doctrines that have been developed in the attempt to answer them, have brought important benefits to the science of mind. The sensations (it was answered) that we are conscious of, and that we receive joined together in a certain uniform manner, imply not only •a law or laws of connection but also •a cause external to our mind; and it’s this cause which by its own laws determines the laws according to which the sensations are connected. The scholastics used to call this external cause by the name I have already used, ‘substratum’; and its attributes (as they put it) inhered in it—literally meaning were stuck in it. To this substratum the name ‘matter’ is usually given in philosophical discussions.† But it was soon accepted by all who thought about the subject that the existence of matter can’t be proved by extrinsic evidence. So the answer usually given to Berkeley and his followers is this:

The belief ·in the existence of matter· is intuitive. Mankind in all ages have felt themselves compelled, by a necessity of their own nature, to regard their sensations as having an external cause. Even those who deny this in theory yield to the necessity in practice: in speech, thought, and feeling they join the man in the street in taking their sensations to be the effects of something external to them. This knowledge is as obviously intuitive as our knowledge of our sensations themselves is intuitive.

And here the question merges into the fundamental problem of metaphysics properly so-called, and I leave it there.

The extreme doctrine of idealist metaphysicians ·such as Berkeley·—that objects are nothing but our sensations and the laws that connect them—has not been generally adopted by subsequent thinkers; but they are generally thought to have been right in one really important part of their doctrine, namely the thesis that all we know of objects is the sensations they give us and the order in which they are given. Kant himself is as explicit on this point as Berkeley or Locke. He was firmly convinced that there exists a universe of ‘things in themselves’ that is distinct from the universe of phenomena, i.e. things as they appear to our senses; and he even introduced a technical term, ‘noumenon’ [see Glossary], to denote •what the thing is in itself as contrasted with •how it is represented in our minds; but he accepted that this representation. . . .is all we know of the object, and that the real nature of the thing is, and by the constitution of our faculties must always remain,. . . .an impenetrable mystery to us. [The present version omits Mill’s mentions of a couple of bits of Kant’s doctrine, and several long footnotes in which he discusses some of his contemporaries’ attitudes to the matter now being discussed.]

There’s not the slightest reason to think that what we call an object’s ‘sensible qualities’ are like anything inherent in the object itself. . . . Just because x causes y, it doesn’t have to be the case that x is like y: an east wind isn’t like the feeling of cold, heat isn’t like the steam of boiling water. So why should matter resemble our sensations? Why should the inmost nature of fire or water resemble the impressions those objects make on our senses? What grounds have we for inferring from the effects anything about the cause except that it is a cause adequate to produce those effects? So we can take it a truth that is obvious in itself and accepted by everyone that we need to listen to, that we don’t and can’t possibly learn anything about the external world except the sensations we experience from it. [A long footnote here discusses the views of some contemporaries—views that Mill hadn’t known about when this book was first written [we are reading its eighth edition]. The main upshot here is Mill’s thesis that the two somewhat opposing doctrines—•that there are no noumena and •that we have intuitive knowledge that there are noumena—are both irrelevant to logic.]

§8. Body having now been defined as the external cause (and, according to the more reasonable opinion, the unknown external cause) to which we refer our sensations, we now have to create a definition of mind. And after what we have just been through, this won’t be hard. For alongside this:

•our conception of a body is of an unknown arousing cause of sensations,

we can say this:

•our conception of a mind is of an unknown recipient of those sensations and indeed of all our other feelings.

Body is the mysterious something which that stimulates the mind to feel, and mind is the mysterious something that feels and thinks. There’s no need for me to give details of a sceptical theory calling in question whether there is any such thing as mind as a thing in itself. . . .; but I do have to say that we are, and unless our faculties change must always remain, entirely in the dark about the innermost nature (whatever ‘innermost nature’ means) of the thinking principle [see Glossary], as well as about the inmost nature of matter. All we’re aware of, even in our own minds, is what James Mill called a certain ‘thread of consciousness’, i.e. a series of sensations, thoughts, emotions, and volitions, more or less numerous and complicated. There’s a something that I call ‘myself’, or ‘my mind’, which I regard as distinct from these sensations, thoughts, etc.; a something that I conceive to be not •the thoughts but •the being that has the thoughts, and which I can conceive as existing forever in a state of quiescence, without any thoughts at all. But although this being is myself, I know nothing about it except its series of states of consciousness. Just as bodies show themselves to me only through the sensations I think they cause, so also the thinking principle (or mind) in my own nature shows itself to me only by the feelings of which it is conscious. All I know about myself are my capacities for feeling, i.e. being conscious (including, of course, thinking and willing); and with my present faculties I can’t conceive of learning anything new about my own nature except learning that I have some additional capacities—not known to me now—for feeling, thinking, or willing.

Summarising: Just as body is the unfeeling item that we are naturally prompted to regard as the cause of a certain portion of our feelings, so mind is the sentient subject of all feelings—it’s what has them or feels them. But according to the best existing doctrines we know nothing about the nature of either body or mind other than the feelings that body causes and mind experiences. And if we do know anything, logic has nothing to do with it or with how that knowledge is acquired.

So much for feelings and substances. Now let us move on to the third and only remaining class of nameable things.

III. ATTRIBUTES, STARTING WITH QUALITIES. . .

§9. From what I have said about substance it’s easy to deduce what should be said about attribute. If we don’t and can’t know anything about bodies but the sensations they cause in us, those sensations must be all that we can—when you come right down to it—mean by their attributes; and the distinction that we verbally make between the properties of things and the sensations we receive from them must originate in the convenience of discourse rather than in the nature of what the words signify. [That is the ‘astonishing’ remark mentioned in the long note here.]

Attributes are usually classified under the headings: Quality, Quantity, and Relation. I shall start with Quality.

Let’s work with a so-called ‘sensible quality’—specifically whiteness. What do we mean when we say that snow has the quality whiteness? Simply that when snow is present to our sense-organs we have a kind of sensation that we customarily call ‘the sensation of white’. How do I know that snow is present? Obviously, by the sensations I derive from it. I infer that the object is present because it gives me a certain group or series of sensations; and when I ascribe to the object the attribute whiteness, all I mean is that one member of that group or series is the sensation that I call ‘the sensation of white colour’.

That’s one view of this subject, but there’s also a different possible view, which goes like this:

It’s true that •we know nothing of sensible objects except the sensations they cause in us; and that •our receiving from snow the particular sensation called ‘a sensation of white’ is our only reason for saying that the snow has the quality whiteness. But because x is the sole evidence of the existence of y it doesn’t follow that x is y. The attribute whiteness is not the fact of receiving the sensation, but something in the object itself, a power inherent in it, something that enables the object to produce the sensation. When we say that snow has the attribute whiteness, we don’t merely say that the presence of snow produces that sensation in us, but that it does so through and by reason of that power or quality.

For the purposes of logic it doesn’t really matter which of these opinions we adopt. The full discussion of the subject belongs to. . . .metaphysics; but I will say this: for the doctrine that there exists a peculiar species of entities called ‘qualities’ I can see no basis except in the human mind’s disposition, whenever we meet with two names that aren’t precisely synonymous, to suppose that they must be the names of two different things; whereas they may be names of one thing viewed in two different lights, or in different surrounding circumstances. Because ‘quality’ and ‘sensation’ aren’t interchangeable in all contexts it is supposed that they can’t both signify the same thing—namely the feeling we’re affected with through our senses by the presence of an object. There’s no absurdity in supposing that this impression or feeling may be called a ‘sensation’ when considered in itself and a ‘quality’ when looked at in relation to some object whose presence to our sense-organs causes that sensations (among others) in our minds. And if this is an admissible supposition, those who contend for a separate entity called a ‘quality’ ought to show that their opinion •is preferable rather than •being merely a lingering remnant of the old doctrine of occult [= ‘hidden’] causes—the exact same absurdity that Molière so nicely ridiculed in his play The Imaginary Invalid when he made one of his pedantic physicians ‘explain’ why opium produces sleep by saying that ‘opium has a soporific virtue’. [This sort of empty ‘explanation’ has been expressed in terms of ‘virtue’, ‘occult cause’, and ‘faculty’. When Mill expresses it in terms of ‘power’, that’s because he thinks, as most philosophers now do, that ‘faculty’ is just a fancy word for ‘power’.]

It’s obvious when the physician said that opium has a ‘soporific virtue’ he merely said again that it produces sleep—he didn’t explain it. Similarly, when we say that snow is white because it has the quality of whiteness we’re only saying again, in more technical language, that it causes in us the sensation of white. But doesn’t the sensation have to have some cause? Yes: its cause is the presence of the group of phenomena that is called the ‘object’. All that we know about the matter is this: Whenever the object is present and our sense-organs are in their normal state, the sensation occurs. After assigning a reliable and intelligible cause, we have no reason to suppose that there is also an occult cause that enables the real cause to produce its effect! Why does the presence of the object cause this sensation in me? I don’t know. I can only say that such is my nature and the nature of the object; that’s just how things are. And we’ll come eventually to this answer even after interpolating the imaginary entity, ·the occult cause·. However many links there are in a causal chain, we’ll still be unable to explain how any one link produces the one next to it. It’s as easy to comprehend that •the object produces the sensation directly and at once as that •it produces it by something else called the ‘power’ of producing it.

But I would have to go far beyond the boundaries of logic to remove all the difficulties that you may feel about this view of the subject, so I’ll settle for the little I have already said, and for the purposes of logic I’ll adopt a language compatible with either view of the nature of qualities. I shall say—what at least admits of no dispute—that the quality of whiteness ascribed to the object snow is grounded on its arousing in us the sensation of white; and. . . .I’ll call the sensation of white the ‘foundation’ of the quality whiteness. For logical purposes the sensation is the only essential part of what is meant by the word—the only part we can ever care about proving. When that is proved, the quality is proved; if an object causes a sensation it does of course have the power of causing it!

. . . THEN RELATIONS. . .

§10. The qualities of a body (I repeat) are the attributes grounded on the sensations aroused in our minds by the presence of that particular body. But when we ascribe to any object x the kind of attribute called a ‘relation’, the foundation of the attribute must be something in which other objects are concerned besides x and the percipient.

If two things can be given correlative names it is proper to say that there is a relation between them. So we may expect to discover what relations in general are if we •list the principal cases in which mankind have imposed correlative names and •see what these cases have in common.

Here are some items to put in the list:

•x is like y

•x is unlike y

•x is near y

•x is far from y

•x is before, after, along with y

•x is greater than, equal to, less than y

•x is the cause of y, the effect of y

•x is the master, servant, child, parent, debtor, creditor, sovereign, subject, attorney, client of y.

What is the common character possessed by everything in this heterogeneous and discordant list?

Resemblance has to be considered separately, so I set it aside for now. One thing that is common to all the rest—and it seems to be the only one—is that in each of them x and y both have roles in some fact or phenomenon which does or did or may be expected to exist or occur. This fact or phenomenon is what the Aristotelian logicians called the foundation of the relation [Mill gives it in Latin, fundamentum relationis]. Thus in the relation of greater and less between two magnitudes x and y, the foundation of the relation is the fact that x could under certain conditions be included in the space occupied by y without entirely filling it. In the relation of master x and servant y, the foundation of the relation is the fact that y has undertaken (or is compelled) to perform certain services for the benefit and at the bidding of x. Examples could be indefinitely multiplied; but it’s already obvious that

•whenever two things are said to be related, there’s some fact or series of facts into which they both enter; and ·conversely· that

•whenever any two things are involved in some one fact or series of facts, we may ascribe to them a mutual relation grounded on that fact or series of facts.

Even if they have nothing in common except what they share with all things, namely that they are members of the universe, we call that a relation and label them as ‘fellow-creatures’, ‘fellow-beings’, or ‘fellow-inhabitants of the universe’. A relation will be more or less specialised, more or less complicated, depending on the nature of the fact on which it is grounded. And there are as many conceivable relations as there are conceivable kinds of fact in which two things can be jointly involved.

So there’s a parallelism here:

•A quality of x is an attribute grounded on the fact that a certain sensation. . . .is produced in us by x;

•A relation between x and y is an attribute grounded on some fact into which x enters jointly with y.

The very same kind of elements make up the two facts— namely states of consciousness. In the case, for example, of any legal relation—debtor and creditor, principal and agent, guardian and ward—the foundation of the relation consists entirely of thoughts, feelings and volitions of the persons themselves or of others who are concerned in the same series of transactions. I mean thoughts etc. that do occur or that would occur if certain conditions existed; for example the intentions a judge •would form if a complaint were made to his court concerning the infringement of any of the legal obligations imposed by the relation; and the acts the judge •would perform as a result of that. (Remember that ‘act’ is just another word for ‘intention followed by an effect’, and that ‘effect’ is just another word for ‘sensations or some other feelings’ caused in the agent himself or in somebody else.) Everything implied by the names that express the relation comes down to states of consciousness; outward objects are supposed throughout as the causes of some of those states of consciousness, and minds are supposed as the subjects by which they are experienced, but the external objects and the minds are known to exist only through states of consciousness.

Relations aren’t all as complicated as those legal ones. The simplest of all relations are those expressed by ‘before’, ‘after’ and ‘simultaneous with’. If we say that dawn came before sunrise, the fact in which dawn and sunrise were jointly involved consisted only of those two things themselves; no third thing entered into the fact or phenomenon. . . . Dawn and sunrise announce themselves to our consciousness by two successive sensations. Our consciousness of the succession of these sensations isn’t a third sensation added to those two. . . . To have two feelings at all implies having them either successively or simultaneously. Two sensations or other feelings being given, they have to be successive or simultaneous—the nature of our faculties insists on that. No-one has been able to analyse the matter any further than that, and there’s no need to try.

§11. It’s somewhat the same with two other sorts of relations, •likeness and •unlikeness. Suppose I have a pair of simple sensations, both of white; and another pair, one of white and one of black. I call the first two sensations ‘alike’, the second two ‘unalike’. The fact or phenomenon on which these relations are grounded consists of •two sensations and •a feeling of resemblance or of lack of resemblance. . . . Resemblance is evidently a feeling, a state of the observer’s consciousness. Whether the feeling of the resemblance of the two sensations is a third state of consciousness. . . .or whether (like the feeling of their succession) it is involved in the sensations themselves, may be a matter of discussion. But either way, these feelings of resemblance and dissimilarity are parts of our nature; and parts so far from being analysable that they are presupposed in every attempt to analyse any of our other feelings. So likeness and unlikeness must—along with beforeness, afterness and simultaneity—stand apart among relations as sui generis. They are attributes grounded on facts, i.e. on states of consciousness, but states that are special, unanalysable and inexplicable.

But though likeness and unlikeness can’t be resolved into anything else, complex cases of them can be resolved into simpler ones. When we say of two things that have parts that they are alike, their likeness can be analysed: it is composed of likenesses between the parts of one and the parts of the other, and of likeness in their arrangement. . . . If one person mimics another with any success, that complex over-all similarity must be made up of ever so many simple likenesses—in a succession of bodily postures, in the accents and intonations of the voice, in the choice of words, and in the thoughts or sentiments expressed. . . .

Every case of likeness or unlikeness that we know about comes down to likeness or unlikeness between states of mind, our own or someone else’s. We say that one body is like another, but all we know of bodies is the sensations they cause us to have; so what we mean really is that there’s a resemblance between some or all of the sensations caused by the two bodies. We say that two attributes are like one another, but all we know of attributes is the sensations or states of feeling they are grounded on; so we mean really that those sensations or states of feeling resemble each other. We say that two relations are alike. A resemblance between relations is sometimes called ‘analogy’ (one of many meanings of that word). The relation of Priam to Hector—namely, father to son—resembles the relation of Philip ·of Macedon· to Alexander ·the Great·; resembles it so closely that they are called the same relation. Cromwell’s relation to England resembles Napoleon’s to France, though not so closely as to be called the same relation. In each case, the meaning must be that there’s a resemblance between the facts that were the foundation of the relation.

Resemblances range from •perfect undistinguishableness to •something extremely slight. [Mill then discusses ‘A thought suggested to the mind of a genius is like a seed cast into the ground’, explaining at some length that this rests on resemblances between relations.]

Hardly anyone is sufficiently on his guard against a certain ambiguity of language concerning relations. When two things are so alike as to be indistinguishable from one another, this likeness is •often called ‘identity’ and the two similar things are said to be ‘the same’. This doesn’t happen •always because we don’t talk like that about similar pairs of visible objects, e.g. two persons; but we always do so when speaking of feelings—‘The sight of that whale carcass gives me the same feeling today that it did yesterday, or the same that it gives to my wife’. This is obviously a misuse of ‘same’, because the feeling I had yesterday is gone, never to return; what I have today is another feeling very like yesterday’s; and it’s obvious that two people can’t be experiencing ‘the same’ feeling in the sense in which they can both sit at ‘the same’ table. The ambiguity is also at work when we say that two persons are ill with ‘the same’ disease, or have ‘the same’ official duties, meaning that their duties are similar. A different sense of ‘same’ is at work when we say that two people are engaged in ‘the same adventure’ or sailing in ‘the same ship’. Able people are often confused and led into bad arguments by their not being properly aware that they (sometimes unavoidably) use the same name to express ideas as different as those of •identity and •indistinguishable resemblance. Whately is almost the only current writer to have called attention to this distinction and to the ambiguity connected with it.

Several other relation-words really stand for resemblance. For example, ‘equality’. This is just another word for exact resemblance in respect of quantity. And this leads into the third and last of the three headings under which attributes are commonly arranged.

. . . AND FINALLY QUANTITY

§12. [In this paragraph, the variables ‘x’, ‘y’ etc. are not Mill’s; nor are the subscripts.] Let us take two things that are exactly alike except in quantity—x is a gallon water and y is 10 gallons of water. Each makes its presence known to us by sensations that it causes us to have; but we don’t mistake x for y, so there must be a likeness_1 between the two sets of sensations. Similarly, a gallon of water z and a gallon of wine w make their presence known by two sets of sensations, and again the sensations are unalike_2. So we have here two cases of unlikeness-between-sensations; and these unlikenesses are themselves unalike_3—we register that fact when we say that one concerns quantity and the other quality. What exactly is this unlikeness_3? It’s not for logic to analyse it, or even to decide whether it can be analysed. . . . All I want here is to show, that when we say of two things that they differ in quantity, just as when we say that they differ in quality, the assertion is always grounded on a difference in the sensations they cause. No-one would deny that seeing or lifting or drinking ten gallons of water involves a set of sensations different from those of seeing or lifting or drinking one gallon. . . . I don’t undertake to say what the difference in the sensations is. Everybody knows, and nobody can tell; any more than any one could tell what white is to a person who had never had the sensation. . . .

SUMMING UP ON ATTRIBUTES

§13. So all the attributes of bodies that are classed under quality or quantity are grounded on the sensations we get from those bodies, and can be defined as the powers that the bodies have of causing those sensations. And we’ve seen that the same general account can be given for most of the attributes usually classed under relation. They too are grounded on some fact into which the related objects enter as parts; that fact has no meaning and no existence to us except the series of states of consciousness by which it makes itself known; and the relation is simply the power the object has of taking part along with the correlated object in causing the series of sensations. We have had to recognise a somewhat different character in certain special relations—succession and simultaneity, likeness and unlikeness. These can’t be analysed in the same way because they aren’t grounded on any fact distinct from the related objects themselves. But these relations are themselves states of consciousness: resemblance is merely our feeling of resemblance; succession is merely our feeling of succession. Or at any rate all that we do or can know about these relations is confined to the relations between our states of consciousness. . . .

§14. In this discussion I have for simplicity’s sake considered bodies only, and omitted minds. But everything I have said is applicable, mutatis mutandis [see Glossary], to minds as well. The attributes of minds are like those of bodies in being grounded on states of consciousness; but in the case of a mind we have to consider its own states as well as the states it causes in other minds. Every attribute of a mind consists either in that mind’s being itself affected in a certain way or its affecting other minds in a certain way. . . .

. . . .A mind doesn’t cause sensations (as a body does), but it may cause thoughts or emotions. The most important attributes ascribed on this ground are terms expressing approval or blame. We say ‘He is admirable’, meaning ‘His mind is admirable’, meaning that our contemplation of his mind arouses admiration in us—with an implication also that we approve of our having that feeling. Sometimes a single attribute is really two combined: ‘He is generous’ means something about his state of mind and also (because it is a term of praise) something about the approval his mind arouses in ours. . . .

We can ascribe attributes to bodies, too, on the ground of •ideas and emotions, and not solely on the ground of •sensations. ‘That’s a beautiful statue’ is grounded on the special feeling of pleasure the statue produces in our minds, and that’s an emotion, not a sensation.

VII. SUMMING UP THIS CHAPTER

§15. My survey of the varieties of things that have been or can be named—that have been or can be •predicated of other things or themselves •made the subject of predications—is now concluded. [Mill now re-states his results—more briefly but with no change of content. After that:] So my analysis gives us the following classification of all nameable things:

(1) Feelings, i.e. states of consciousness.

(2) The minds that experience those feelings.

(3) The bodies, i.e. external objects that arouse certain of those feelings, together with the powers by which they arouse them.

(4) The successions and co-existences, likenesses and unlikenesses between feelings or states of consciousness.

. . . .I don’t think that a sound philosophy would support the inclusion of ‘powers’ in (3), but I put them in because their existence is taken for granted in common language, and it wouldn’t be prudent for me to deviate from that.

Until something better can be suggested, I offer this as a substitute for Aristotle’s Categories, considered as a classification of existences. The practical application of it will appear when I start looking into the import of propositions—i.e. into what the mind actually believes when it gives its ‘assent’ to a proposition.

If my classification is correct, the above four classes comprise all nameable things; so of course some or all of them must •compose the signification of all names and •make up every fact.

•Facts composed solely of feelings or states of consciousness are often called •psychological or •subjective facts.

•Facts composed at least partly of. . . .substances and attributes are called •objective facts.

So we can say that every objective fact is grounded on a corresponding subjective one, and apart from that it has no meaning to us except as a name for the unknown and inscrutable process by which that subjective or psychological fact is made to happen.

Chapter 4. Propositions

§1. With propositions as with names, I have to start with some fairly elementary points about their form and varieties, before entering into the analysis of the import conveyed by them, which is the real subject and purpose of this preliminary Book.

A proposition, I repeat, is a bit of discourse in which a predicate is affirmed or denied of a subject. A predicate and a subject are all we need to make a proposition; but merely putting two names side by side doesn’t tell anyone that they are a predicate and a subject, i.e. that one is intended to be affirmed or denied of the other; so there has to be some way of indicating that that is the intention—some sign to distinguish a predication from any other kind of discourse. This is sometimes done by an inflection of one of the words, as when we say ‘Fire burns’—the ‘s’ in ‘burns’ shows that we mean to affirm the predicate burn of the subject fire. But it is more often done by ‘is’ (for affirmation) or ‘is not’ (for negation), or by some other part of the verb ‘to be’. The word that serves as a sign of predication is called the ‘copula’. It’s important to think clearly about what the copula is and does: confused notions about this have helped to spread mysticism over the field of logic and perverted its speculations into wars over words.

Some writers have thought that the copula is more than a mere sign of predication—that it also signifies existence. The proposition ‘Socrates is just’ may seem to imply not only •that ‘just’ can be affirmed of Socrates but also •that Socrates is, i.e. that he exists. But this only shows that ‘is’ is ambiguous: as well as doing the work of a copula in affirmations, it also has a meaning of its own which lets it be the predicate of a proposition. The two don’t have to go together: from ‘A centaur is a fiction of the poets’ you can’t infer that a centaur exists, because the proposition explicitly denies this.

Books could be filled with the frivolous speculations about the nature of Being (. . . ‘ens’, ‘entitas’, ‘essentia’ and the like) that have arisen from overlooking this double meaning of ‘be’, supposing that when it signifies to exist it must basically answer to the same idea as it does when it signifies to be F for some specific value of F—

•to be a man,

•to be Socrates,

•to be seen or spoken of,

•to be a phantom,

•to be a nonentity

—having one meaning that fits all these cases! The fog rising from this narrow spot spread in ancient times over the whole surface of metaphysics. But we shouldn’t put ourselves above the great intellects of Plato and Aristotle because we can now avoid many errors that they, perhaps inevitably, committed. . . . The Greeks seldom knew any language but their own, and that made it much harder for them than it is for us to detect ambiguities. One advantage of knowing several languages—especially ones in which eminent thinkers have expressed their thoughts—is the lesson regarding ambiguity that we learn by finding that a single word in one language corresponds in different contexts to different words in another. Without that help, even the strongest understandings find it hard to believe that things that share a name don’t also in some way share a nature. . . . But once the habit ·of ambiguity-spotting· has been formed, one doesn’t need high intelligence to detect ambiguities, even ones that are common to many languages; so it’s surprising that the ambiguity of ‘to be’, though it exists in modern languages as well as in ancient ones, should have been overlooked by almost everyone. The quantity of futile speculation arising from a misunderstanding of the nature of the copula was hinted at by Hobbes; but I think it was James Mill who first clearly described the ambiguity, and pointed out how many errors in the accepted systems of philosophy it has been responsible for. . . .

I shall now briefly review the principal distinctions among propositions, and the technical terms most commonly used to express them.

§2. A proposition being a portion of discourse in which something is affirmed or denied of something, the first division of propositions is into affirmative and negative. An affirmative proposition is one where the predicate is affirmed of the subject (‘Caesar is dead’); a negative proposition is one where the predicate is denied of the subject (‘Caesar is not dead’). The copula in the negation is ‘is not’; in the affirmation it is ‘is’.

Hobbes and some other logicians have stated this distinction differently, recognising only one form of copula, ‘is’, and attaching the negative sign to the predicate. According to these writers, ‘Caesar is dead’ and ‘Caesar is not dead’ have the same copula but different predicates, ‘dead’ and ‘not dead’; so they define a negative proposition as one in which the predicate is a negative name. . . . These writers had the idea that they could get rid of the affirming/denying distinction by treating every case of denying as the affirming of a negative name. But what is a negative name? A name expressing the absence of an attribute; so when we affirm a negative name what we’re predicating is absence, not presence; we’re asserting not that anything is but that something is not; and the best name for that operation seems to ‘denying’. The fundamental distinction is between

•a fact and the non-existence of that fact;

•seeing something and not seeing it,

•Caesar’s being dead and his not being dead.

If this were a merely verbal distinction, the thesis that brings both within the same form of assertion would be a real simplification; but the distinction is real—it’s in the facts—and what deserves the label ‘merely verbal’ is not this distinction but the thesis that smudges it by treating the difference between two kinds of truths as a mere difference between two kinds of words. . . .

Something like that also applies to most of the distinctions among propositions that are said to concern their modality—e.g. difference of tense or time: ‘The sun did rise’, ‘ The sun is rising’, ‘The sun will rise’. These differences could also be glossed over by regarding the temporal element as a mere detail about the predicate: ‘The sun is an object having risen’, ‘The sun is an object now rising’, ‘The sun is an object to rise hereafter’. But this simplification would be merely verbal. Past, present and future don’t constitute different kinds of rising; they are designations belonging to the event asserted, to the sun’s rising today. What they act on is not •the predicate but •the applicability of the predicate to that particular subject. What we affirm to be past, present or future is not

•what the subject signifies, or

•what the predicate signifies, but

•what the predication signifies

—something expressed by the whole proposition and not by either or both of the terms. So the temporal element is properly considered as attaching to •the copula, which is the sign of predication, and not to •the predicate. Nothing like this applies to ‘Caesar may be dead’, ‘Caesar is perhaps dead’, ‘It is possible that Caesar is dead’, because these are really assertions not of •anything relating to the fact itself but of •the state of our mind in regard to it—e.g. ‘Caesar may be dead’ means ‘I am not sure that Caesar is alive’.

§3 . The next division of propositions is into simple and compound or complex. A simple proposition is one in which one predicate is affirmed or denied of one subject. A compound proposition has more than one predicate, or more than one subject, or both.

At first glance this division looks absurd—a solemn distinction of things into •one and •more than one, like dividing horses into •single horses and •teams of horses. It’s true that what is called a compound proposition is often not one proposition but several, held together by a conjunction. ‘Caesar is dead and Brutus is alive’ or even ‘Caesar is dead but Brutus is alive’—in each of these there are two assertions, and calling them a complex proposition is like calling a street a complex house. The words ‘and’ and ‘but’ do have a meaning; but that meaning, far from making the two propositions one, adds a third proposition to them. All particles are abbreviations, and usually abbreviations of propositions, a kind of short-hand which quickly suggests to the mind something that it would take a proposition or a series of propositions to express fully. For example, ‘Caesar is dead and Brutus is alive’ is equivalent to: ‘Caesar is dead; Brutus is alive; it is desired that those two propositions be thought of together’. As for ‘Caesar is dead but Brutus is alive’—that means the same as the other with a fourth proposition, namely ‘Between the first two propositions there’s a contrast’. . . .

In those examples the two propositions are kept visibly distinct, each subject having its own predicate and vice versa. Often for brevity’s sake the propositions are blended together: ‘Peter and James preached at Jerusalem and in Galilee’ contains four: ‘Peter preached at Jerusalem’, etc. . . .

We have seen that when the two or more propositions contained in a ‘complex proposition’ are stated outright and unconditionally, it’s not a proposition but a plurality of propositions; and if the whole thing is true then so is each of its constituent propositions taken separately. But there’s a kind of proposition which, though it. . . . does in a sense consist of several propositions, contains only one assertion; and its truth doesn’t imply that each of the simple propositions that make it up is also true. For example:

•A is B or C is D;

•A is B if C is D.

The former proposition is called ‘disjunctive’, the latter ‘conditional’; ‘hypothetical’ used to be common to both.

Whately and others have pointed out that the disjunctive form is resolvable into the conditional, because every disjunctive proposition is equivalent to two or more conditional ones. ‘Either A is B or C is D’ means ‘if A is not B, C is D; and if C is not D, A is B’. So all hypothetical propositions are disjunctive in form but conditional in meaning; and ‘hypothetical’ and ‘conditional’ may be and generally are used synonymously. Logicians use ‘categorical’ as their name for propositions in which the assertion [see Glossary] doesn’t depend on a condition.

A hypothetical proposition is not. . . .a mere cluster of simple propositions. The simple propositions that enter into it are not part of the assertion it conveys. When we say ‘If the Koran comes from God, Mohammed is the prophet of God’, we don’t intend to affirm either that the Koran comes from God or that Mohammed is his prophet. The hypothetical proposition may be indisputably true even if each of those simple propositions is false. What is asserted isn’t that either of them is true but that the second of them can be inferred from the first. Then what is the subject of the hypothetical proposition and what is its predicate?. . . . The subject is the proposition ‘Mohammed is the prophet of God’, and what is affirmed of it is that this is a legitimate inference from the proposition ‘The Koran comes from God’. The subject and predicate of a hypothetical proposition are thus names of propositions. The subject is some one proposition. The predicate is a general relative name applicable to propositions, of the form ‘. . . an inference from so and so’. We see again here that particles are abbreviations: ‘If A is B, C is D’ turns out to be an abbreviation of ‘The proposition “C is D” is a legitimate inference from the proposition “A is B”’.

So hypothetical propositions aren’t as different from categorical ones as they at first seemed to be. In each of them one predicate is affirmed of just one subject; but a conditional proposition is about a proposition; the subject of the assertion is itself an assertion. . . . And there are other kinds of assertions about propositions. . . .because other things can be predicated of a proposition:

•‘The whole is greater than its part’ is an axiom in mathematics.

•‘The Holy Ghost proceeds from the Father alone’ is a tenet of the Greek Church.

•The doctrine of the divine right of kings was renounced by Parliament at the Revolution.

•‘The Pope is infallible’ has no support in Scripture.

In each of these the subject of the predication is an entire proposition. . . .

Given that •hypothetical propositions differ from others much less than one might think, judging from their form, isn’t it surprising that •they loom so large in treatises on logic? No! To see why, remember that what they predicate of a proposition—namely that it can be inferred from something else—is precisely the one of its attributes that a logician is which most of all concerned with.

§4. The next common division of propositions is into universal, particular, indefinite, and singular—a classification based on how generally the subject term is to be understood. Here are examples:

•‘All men are mortal’—universal.

•‘Some men are mortal’—particular.

•‘Man is mortal’—indefinite.

•‘Julius Caesar’ is mortal—singular.

A proposition is singular if its subject is an individual name, which doesn’t have to be a proper name. ‘The founder of Christianity was crucified’ is as much a singular proposition as ‘Christ was crucified’.

When the subject term is a general name, we may intend to affirm or deny the predicate of •all the things it denotes or only of •some of them. When the predicate is affirmed or denied of •each of the things denoted by the subject, the proposition is universal; when it is affirmed or denied of •only some undefined portion of them, it is particular. So these are all universal:

•‘All men are mortal’,

•‘Every man is mortal’,

•‘No man is immortal.

The last one is a universal proposition because its predicate, ‘immortal’, is denied of each and every individual denoted by the term ‘man’. It could instead have been expressed as ‘Every man is not-immortal’. But ‘Some men are wise’ and ‘Some men are not wise’ are particular propositions, because the predicate ‘wise’ is affirmed (denied) not of each and every man but only of each and every man in some portion of men, without saying what portion. If the portion were specified, the proposition would become either •singular or •universal with a different subject—e.g. ‘All properly instructed men are wise’. There are other forms of particular propositions, such as ‘Most men are imperfectly educated’: it doesn’t matter how large a portion of men the predicate is asserted of, as long as it isn’t specified how that portion is marked off from the rest.

[A so-called ‘indefinite’ proposition, Mill says, is one that doesn’t make clear whether its it is an assertion about all the things denoted by subject term or only some of them. He agrees with Whately that this is a blunder, like that of the grammarians who say that there are three grammatical genders—masculine, feminine, and doubtful. What we really have here is not a kind of proposition but a mere unclarity or vagueness, which indeed is often removed by the context. He follows Bain in emphasising ‘indefinite’ propositions with names of materials as their subjects: ‘Food is chemically constituted by carbon, oxygen, etc.’ is meant as a universal proposition, whereas ‘Food is necessary to animal life’ is meant as a particular.]

[In two further paragraphs Mill introduces a time-hallowed technical term, namely ‘distributed’. In these:

•‘All A are B’

•‘Some C are D’

the term A stands for is distributed, the other three are undistributed. Mill says that this distinction helps a lot in stating and proving rules of the syllogism, but in fact he doesn’t use it when gets to that topic in Book II.]

There are many more ways of classifying propositions than those I have presented, some of them quite important. I’ll explain and illustrate them when suitable occasions arise.

Chapter 5. The import of propositions

§1. An inquiry into the nature of propositions must aim either to analyse (i) the state of mind called ‘belief’, or to analyse (ii) what is believed. All languages distinguish (ii) a doctrine or opinion from (i) the fact of entertaining the opinion; (i) assent from (ii) what is assented to.

Logic as I understand the term has no concern with the nature of the act of judging or believing; the consideration of that mental act belongs to another science. Yet philosophers from Descartes downward, and especially from the era of Leibniz and Locke, haven’t observed this distinction and would have dismissed out of hand any attempt to analyse the import of propositions that wasn’t based on an analysis of the act of judgment. They would have said: ‘A proposition is merely the verbal expression of a judgment. What matters is the thing expressed, not the mere expression of it. When the mind assents to a proposition, it judges. When we find out what the mind does when it judges, we’ll know what propositions mean; there’s no other way.’

In line with these views, almost all the writers on logic in the last two centuries—English, German, or French—made their theory of •propositions nothing but a theory of judgments. They considered a proposition (or a judgment—they used the two words indiscriminately) to consist in affirming or denying one idea of another. To judge was to

•put two ideas together, or to

•bring one idea under another, or to

•compare two ideas, or to

•perceive the disagreement between two ideas’

Or replace ideas by conceptions or whatever else the writer preferred as an all-purpose name for mental representations. And these writers held that such ideas are essentially the subject-matter and substance of the operations of judging and reasoning. . . .

Whenever we make a judgment, e.g. judging that gold is yellow, something happens in our minds that is partly described by one or other of these theories. We must have the ideas of gold and of yellow, and these must be brought together in our mind. But there’s obviously more to it than that, because we can put two ideas together without any act of belief, for example when we merely imagine a golden mountain, or when we disbelieve—for you can’t disbelieve that Mohammed was an apostle of God without putting together your ideas of Mohammed and of apostle of God. What does happen in assent or dissent besides putting two ideas together? That is an intricate metaphysical problem; but the solution of it—whatever it turns out to be—can’t have anything to do with the import of propositions. Why not? Because propositions are hardly ever assertions about our ideas of things; they’re assertions about the things themselves. To believe that gold is yellow I must have the ideas of gold and of yellow, and something involving those ideas must take place in my mind; but my belief isn’t about the internal ideas—it’s about external gold and yellow, about gold and the effect it has on human sense-organs. . . . It’s true that if I am to believe this external fact another fact must take place in my mind, a process must be performed upon my ideas; but so it must in everything else that I do. I can’t dig the ground unless I have the ideas of •the ground and •a spade and so on, and unless I put those ideas together. But it would be ridiculous to say that digging the ground is putting one idea into another! Digging is an operation on the things themselves, though it can’t be done unless I have in my mind the ideas of them. Similarly, believing is an act that has for its subject the facts themselves, but I can’t perform it without already having in my mind a mental conception of those facts. When I say ‘Fire causes heat’ I mean that the natural phenomenon fire causes the natural phenomenon heat. When I mean to say something about ideas I call them ‘ideas’—e.g. ‘A child’s idea of a battle is unlike the reality’. . . .

One of the most fatal errors ever introduced into the philosophy of logic, it seems to me, was the notion that what primarily matters to the logician in a proposition is the relation between the two ideas corresponding to the subject and predicate;. . . .and it’s the main reason why the theory of logic has made so little progress during the last two centuries. The treatises on logic and related branches of philosophy that have been produced since this cardinal [see Glossary] error pushed in, though sometimes written by extraordinarily able men, have nearly all tacitly implied that investigating truth consists in contemplating and handling our ideas. . . ., which amounts to the assertion that the only way to get knowledge of nature is to study it at second hand, as represented in our own minds. [Mill continues with this theme, talking about the ‘great and fruitful truths on most important subjects’ that have been discovered with no help from the logicians, so that natural scientists have come to regard logic as ‘futile’.]

What we must investigate here is not Judgment but judgments; not the act of believing but the thing believed. What is the immediate object of belief in a proposition? What matter of fact does it signify? When I assert a proposition what am I giving my assent to and inviting others to give theirs? What is expressed by the kind of discourse called ‘a proposition’? What is the fact conformity to which makes the proposition true?

§2. One of the clearest and most coherent thinkers the world has produced—I mean Hobbes—has answered this question as follows (·this is not a quotation·):

In every proposition what is signified is the speaker’s belief that the predicate is a name of the same thing that the subject is a name of. If it really is so, the proposition is true. Thus the proposition ‘All men are living beings’ is true, because ‘living being’ is a name of everything of which ‘man’ is a name. ‘All men are six feet tall’ is not true because ‘six feet tall’ is not a name of everything of which ‘man’ is a name.

What this presents as the definition of a true proposition is certainly a property that all true propositions have. If the subject and predicate were names of different things, one name couldn’t be predicated of the other. If it’s true that some men are copper-coloured, it must be true—and the proposition does really assert—that among the individuals denoted by ‘man’ there are some who are also among those denoted ‘copper-coloured’. If it’s true that all oxen ruminate, it must be true that all the individuals denoted by ‘ox’ are also among those denoted by ‘ruminating’; and whoever asserts that all oxen ruminate undoubtedly does assert that this relation holds between the two names.

[Mill now says—with elaborations that are clever but not very helpful—that Hobbes’s account is true of every proposition but is the whole truth about only a few. He continues:] The only propositions of which Hobbes’s principle is a sufficient account are those in which the predicate and the subject are both proper names. Proper names have strictly no meaning; they are mere marks for individual objects; and when a proper name is predicated of another proper name all that’s conveyed is that both names are marks for one object. Thus, what Hobbes offers as a theory of predication in general is a full account of the likes of these:

•‘Hyde was Clarendon.’

•‘Tully is Cicero.’

But it’s a sadly inadequate theory of most propositions. How could Hobbes have thought otherwise? It must because he like other nominalists pretty much ignored words’ connotations and looked for their meaning exclusively in what they denote;. . . .as if the only difference between a proper name and a general name were that the former denotes only one individual and the latter denotes more than one.

But we have seen that the meanings of all names except proper names and some abstract names resides in their connotations. So when we are analysing the meaning of any proposition in which either or both of the predicate and the subject are connotative names, we must look to their connotations and not to what they denote. . . . When Hobbes said that (for instance) ‘Socrates is wise’ is a true proposition because ‘Socrates’ and ‘wise’ are names of the same person, it’s astonishing that such a powerful thinker didn’t ask himself ‘How did they come to be names of the same person?’. ·Well, how did they·? Surely not through the intention of those who invented the words: when mankind fixed the meaning of ‘wise’ they weren’t thinking of Socrates; and when Socrates’ parents gave him that name they weren’t thinking of wisdom! The names happen to fit one person because of a certain fact, that wasn’t known or didn’t exist when the names were invented. And the clue to the fact is in the connotation of the names.

. . . .If in our experience the attributes connoted by ‘man’ are always accompanied by the attribute connoted by ‘mortal’, it will follow that the class man will be wholly included in the class mortal, and that ‘mortal’ will be a name of all things of which ‘man’ is a name: but why? Those objects are brought under each name by having the attributes connoted by it; and •their having those attributes is the real condition that makes the proposition true, not •their being called by the name. Connotative names do not precede but follow the attributes they connote. . . . The most ingenious and refined analysis of the significations of ‘diamond’ and ‘combustible’ couldn’t have shown men that diamonds are combustible. That was discovered by a very different process, namely finding empirically that the attribute of combustibility existed in the diamonds on which the experiment was tried; the number or character of the experiments lending support to the inference that what was true of those individuals is true of all substances ‘called by the name’, that is, of all substances having the attributes the name connotes. . . .

§3. Hobbes’s theory of predication, stated as he stated it, hasn’t been favourably received by subsequent thinkers; yet a theory virtually identical with it—though much less clearly expressed—has almost achieved the status of an established opinion. The prevailing notion of predication these days goes like this:

Predicating A of x consists in putting x into a class— whether x is an individual or itself a class. Thus, ‘Man is mortal’ asserts that the class man is included in the class mortal. ‘Plato is a philosopher’ asserts that the individual Plato is one of those who compose the class philosopher. If the proposition is negative then it excludes something from a class.

This differs only verbally from Hobbes’s account. A class is nothing but an indefinite number of individuals denoted by a general name. What makes them a class is their common name. To refer something to a class, therefore, is to regard it as one of the things that are to be called by that common name. To exclude it from a class is to say that the common name is not applicable to it.

The widespread influence of this view of predication can be seen in the fact that it’s the basis of the celebrated dictum de omni et nullo [Latin = ‘the everything-and-nothing principle’], which says that what is true of a class is true of all things that belong to it. When this is laid down by almost all logicians as the ultimate principle to which all reasoning owes its validity, it’s clear that logicians generally assume that the propositions of which reasonings are composed can only express the process of dividing things into classes and putting everything in its proper class.

This theory seems to me to be a striking example of a common logical error,. . . .namely explaining a thing by something that presupposes it. When I say that snow is white, I ought to be thinking of snow as a class because I’m asserting a proposition about all snow: but I’m certainly not thinking of white objects as a class. The only white object I’m thinking about is snow, and what I am thinking about it concerns only the sensation of white that it gives me. It’s true that when I have judged that snow is white and that various other things are also white, I begin to think of white objects as a class. . . . But this is a conception that •followed those judgments, so it can’t •explain them. Instead of explaining the effect by the cause, this doctrine explains the cause by the effect—I think it is based on a hidden misunderstanding of the nature of classification.

In discussions of these matters, some people write as though they thought this:

Classification is a grouping of definite and known individuals. When names were imposed, mankind

•took into consideration all the individual objects in the universe,

•distributed them into parcels or lists,

•gave the objects in each list a common name, and

•repeated this operation until they had invented all the general names of which language consists.

And now that this has been done, if a question arises about whether a certain general name N can be truly predicated of a certain particular object x, we have only (as it were) to read the roll of the objects on which N has been conferred, and see whether x is among them. The makers of language have predetermined all the objects that are to be in each class, and we have only to consult the record of that previous decision.

No-one will accept this absurd doctrine when thus nakedly stated; but if the commonly accepted explanations of classification and naming don’t imply this theory, it needs to be shown how they can be reconciled with any other.

General names aren’t marks put on definite objects; classes aren’t made by drawing a line round a given number of assignable individuals. The objects composing any class are perpetually fluctuating. We can frame [see Glossary] a class without knowing all—indeed without knowing any—of the individuals in it; we can do this while believing that no such individuals exist. If the meaning of a general name is to consist of the things it is the name of, no general name, except by accident, keeps the same meaning for any length of time. The only way a general name has a definite ·and durable· meaning is by being a name of an indefinite variety of things—namely all the things, known or unknown, past, present, or future, that have certain definite attributes. When we discover empirically that these attributes are possessed by some object not previously known to possess them (as when chemists found that the diamond was combustible), we include this new object in the class; but it did not already belong to the class. We place the individual in the class because the proposition is true; the proposition is not true because the object is placed in the class.[1] [The clause ‘it did not already belong in the class’ is verbatim from Mill. It implies that nothing is in a class until we put it there.]

When ·in Book II· I discuss reasoning, you’ll see how greatly the theory of reasoning has been spoiled by the influence of these erroneous notions, and by the habit of assimilating •all the operations of the human understanding that have truth for their object to •processes of mere classification and naming. Unfortunately, the minds that have been entangled in this net are precisely the ones that have escaped the other big error discussed at the start of this chapter. Since the revolution that dislodged Aristotle from the schools, logicians can almost be divided into •those who have looked on reasoning as essentially an affair of ideas and •those who have looked on it as essentially an affair of names.

Hobbes’s theory of predication. . . .makes truth and falsity completely arbitrary, with no standard but the will of men, and Hobbes acknowledged this.[2] But don’t think that Hobbes, or any of the others who mainly agreed with him about this, did in fact regard the distinction between truth and error as less real or less important than other people: their writings show otherwise—their doctrine didn’t have a strong grip on their minds. No-one has ever really thought that truth is merely propriety of expression, using language in conformity to a previous convention. When they came to particular cases it has always been accepted that verbal questions are different from real ones; that some false propositions are uttered not from ignorance of the meaning of •words but from a misapprehension of •things; that a person who doesn’t have a language can still form propositions mentally, and that they may be untrue. . . . This last admission can’t be made more strongly than it is by Hobbes himself,[3] though he won’t allow such erroneous belief to be called ‘falsity’, but only ‘error’. Hobbes says elsewhere that general names are given to things because of their attributes, which are named by abstract names: ‘Abstract is that which in any subject denotes the cause of the concrete name. . . .’ It is strange that he didn’t see that what he calls the cause of the concrete name is really the meaning of it; and that when we predicate of any subject a name that is given because of an attribute. . . . our intention is not to •affirm the name, but by means of the name •to affirm the attribute.

§4. [Mill now says all this again in application to a proposition of the type ‘[Proper name] is [adjective]’, emphasising that someone who says this is thinking about the •thing named by the •proper name, not about the name itself. Then with a proposition of the form ‘[General name] are [adjective]’: here again the thought is not about the general name but about the things denoted by it.]

With these two things in mind—

•Every attribute is grounded on some fact or phenomenon of outward sense or inward consciousness.

•To say that ‘x has attribute A’ is just one way of saying that x is the cause of (or a part of) the fact or phenomenon upon which A is grounded,

—we can add one more step to complete the analysis. The proposition that one attribute always accompanies another attribute really says that one phenomenon always accompanies another phenomenon. . . . In the proposition ‘All men are mortal’ the word ‘man’ connotes the attributes we ascribe to a certain kind of living creatures on the ground of certain phenomena—physical and mental—that they exhibit. . . . And when we say ‘Man is mortal’ we mean that wherever these physical and mental phenomena are all found, we can be sure that the physical and mental phenomenon called ‘death’ will follow; the proposition doesn’t say when. . . .

§5. We have gone far enough not only to show that Hobbes is wrong but to ascertain what the real import is of the most numerous class of propositions. The object of belief in a proposition. . . .is generally either the •co-existence or the •sequence of two phenomena. At the very start of this inquiry we saw that every act of belief requires two things, and we know now that in most cases these two things are two phenomena, i.e. two states of consciousness; and what the proposition affirms (or denies) regarding them is either that one follows the other or that they co-exist. And this is true of countless cases that no-one would, unprompted, think of describing in those terms. ‘A generous person is worthy of honour’—who would see this as asserting co-existence between phenomena? Yet that’s what it is. A person is termed ‘generous’ because of states of his mind and facts about his conduct; both are phenomena—facts of internal consciousness and physical facts or perceptions of the senses. A similar analysis holds for ‘worthy of honour’: as used here, ‘honour’ means a state of approving and admiring emotion, followed sometimes by corresponding outward acts. ‘Worthy of honour’ connotes all this, together with our approval of the act of showing honour; and all these are phenomena—states of internal consciousness accompanied or followed by physical facts. When we say ‘A generous person is worthy of honour’ we’re affirming co-existence between the two complicated phenomena connoted by the two terms respectively. . . .

After my analysis of the import of names, I needn’t give many examples to illustrate the import of propositions. When there’s any obscurity or difficulty, it comes from the meaning not of the proposition but of the names that compose it—the complicated connotation of many words, the vast series of facts that often constitute the phenomenon connoted by a name. But when we see what the phenomenon is, it’s usually easy for us to seeing what the assertion conveyed by the proposition is, namely the co-existence or succession of the phenomena in question. . . .

This is the •most common meaning propositions are intended to convey, but it’s not the •only one. Sequences and co-existences are asserted not only about phenomena but also about those hidden causes of phenomena—substances and attributes. A substance, though, is nothing but either that which •causes phenomena or that which •is conscious of them; and the same is true mutatis mutandis of attributes. So no meaningful assertion can be made about these unknown and unknowable entities except in virtue of the phenomena by which alone they show themselves to us. When we say ‘Socrates was contemporary with the Peloponnesian war’, the foundation of this assertion, as of all assertions about substances, is an assertion about the phenomena they exhibit, namely this:

The series of facts by which Socrates manifested himself to mankind, and the series of mental states that constituted his sentient existence, went on simultaneously with the series of facts known by the name of ‘the Peloponnesian war’.

[Regarding Mill’s speaking of one assertion as being based on another assertion, see Glossary.] Still, that isn’t all that the proposition (as commonly understood) says; it asserts that the thing in itself, the noumenon [see Glossary] Socrates, was existing and doing or experiencing those various facts during the same time. Co-existence and sequence can be affirmed or denied not only between phenomena but also between noumena, or between a noumenon and phenomena. And both of noumena and of phenomena we may affirm simple existence. But what is a noumenon? An unknown cause. In affirming the existence of a noumenon, therefore, we are affirming causation. So here are two additional kinds of fact that can be asserted in a proposition. Besides the propositions asserting •Sequence or •Co-existence, there are some that assert simple •Existence; and others assert •Causation. The status of this last as a special kind of assertion will come up for reconsideration in Book III.[4]

§6. To these four kinds of matter-of-fact assertion I must add a fifth, Resemblance. (Here I found it impossible to analyse this species of attribute; the only grounding that could be assigned for it was the ·resembling· objects themselves.) Sample propositions involving resemblance: ‘ This colour is like that colour’; ‘The heat of today is equal to the heat of yesterday’. [Mill sketches and shoots down a suggested analysis of resemblance in terms of sequence, and concludes:] Resemblance between two phenomena is more intelligible in itself than any explanation could make it, and in any classification it must be distinguished from ordinary cases of sequence and co-existence.

It is sometimes said that any proposition with a general name as predicate does in fact affirm or deny resemblance. Here is why:

All such propositions affirm that a thing belongs to a class; and things are classed together according to resemblance; so each thing is classed with the things it is thought to resemble most. Thus, when we affirm that gold is a metal we mean that gold resembles other metals more closely than it resembles objects outside that class.

This has some foundation, but not much. The arrangement of things into classes, such as the class metal, is indeed grounded on a resemblance among the things placed in that class, but not on a mere general resemblance; it is grounded on fact that all those things have certain common features, and it is these that ‘metal’ connotes. What the proposition asserts, then, is not a general resemblance but the sharing of these special features. When I say ‘Gold is a metal’ I do imply that if there are any other metals gold must resemble them; but if there were no other metals I could still assert ‘Gold is a metal’, meaning that gold has the various properties implied in the word ‘metal’—which is just what it does mean. Thus, the situation regarding propositions in which objects are assigned to a class because they have the attributes constituting the class is not •that they assert nothing but resemblance but rather that strictly speaking they don’t assert resemblance at all! [Mill in this paragraph ran ‘Gold is a metal’ in harness with ‘Socrates is a man’. One was enough.]

There are, however, two kinds of exception to this. (1) It is sometimes convenient to enlarge a class so as to include things that lack some of its characteristic properties, or have them only in a low degree, provided they resemble that class more than any other, so that propositions that are true of the class will be nearer to being true of those things than any other equally general propositions. For instance, some substances that have very few of the properties by which metals are commonly recognised are nevertheless called ‘metals’; and almost every large grouping of plants or animals has a few anomalous groups on its borders, which are admitted into it by a sort of courtesy. . . . When the class-name is predicated of any group of this description, we are affirming resemblance and nothing more. . . . If my account of predication is to be scrupulously correct it ought to have an added clause to cover predications on such borderline groups, but there’s no need for that complication. It’s not often that such a predication is made on a borderline group, and when it is there’s usually some slight difference in the wording—‘This group is considered as. . . , may be ranked as. . . ’ and so on. . . . (Reasons for admitting borderlines into classes will be more fully discussed in Book IV chapter 7.)

(2) The second kind of exception concerns the classes into which our simple sensations, or other simple feelings, are divided. Sensations of white are classed together not because we can take them to pieces and say ‘They’re alike in this, and not alike in that’, but because we feel them to be alike altogether, though in different degrees. So when I say ‘The colour I saw yesterday was a white colour’ or ‘The sensation I feel is one of tightness’, the attribute I affirm of the colour or of the other sensation is mere resemblance— simple likeness to previous sensations of mine that have been given those names. The names of feelings, like other concrete general names, are connotative; but they connote a mere resemblance. . . .

Existence, Co-existence, Sequence, Causation, Resemblance: one or other of these is asserted (or denied) in every proposition that isn’t merely verbal. This five-fold division covers all matters of fact, all things that can be believed or proposed for belief, all questions that can be asked, all answers that can be given to them.

Bain distinguishes two kinds of propositions of coexistence. ‘In one kind, account is taken of place; they are propositions of order in place.’ He calls the other kind co-inherence of attributes:

‘This is a distinct variety of propositions of coexistence. Instead of an arrangement in place. . . .we have the concurrence of two or more attributes or powers in the same part or locality. A mass of gold contains in every atom the concurring attributes that mark the substance—weight, hardness, colour, lustre, etc. An animal, besides having parts situated in places, has co-inhering functions in the same parts, exerted by the very same masses and molecules of its substance. . . The mind isn’t the source of any propositions of order in place, but it has co-inhering functions. We affirm mind to contain feeling, will, and thought, not in local separation but in commingling exercise. Affirmations of co-inherence are true of the concurring properties of minerals, of plants, and of the bodily and the mental structure of animals.’

This distinction is real and important. ·but it isn’t basic in the way Bain thinks it is·. As I have pointed out, an attribute (except when it’s a simple unanalysable resemblance between the subject and some other things) consists in causing impressions of some sort on consciousness. So the co-inherence of two attributes is merely the co-existence of two states of consciousness. . . . With one complication: this co-existence is sometimes only potential, because an attribute may be regarded as •actually in existence though the fact on which it is grounded is only •potentially present. It’s convenient to regard snow as white even in the dark, because. . . .we’ll be conscious of the whiteness at daybreak. So co-inherence of attributes is still a mildly complex case of co-existence of states of consciousness. But it is of course a totally different thing from order in place; it’s a matter of something’s having two attributes at the same time.

We may sometimes find it convenient, then, to replace

•co-existence and sequence,

by the more specific

•order in place and order in time.

Order in place is one kind of co-existence, a kind that we needn’t analyse any further here. And order in time includes both sequence and the co-existence of attributes, which I have already analysed.

§7. I have thought that I needed to analyse only propositions in which the predicate is a concrete term. But in doing this I have indirectly analysed those in which the terms are abstract. What distinguishes an abstract term from its corresponding concrete isn’t any difference in what they signify; for the real signification of a concrete general name is (I repeat) its connotation; and what the concrete term connotes is the entire meaning of the abstract name. Since everything in the import of an abstract name is also in the import of the corresponding concrete, it’s natural to suppose whatever is in the import of an abstract-terms proposition can also be expressed in a concrete-terms one.

And this turns out to be right. An abstract name is the name of an attribute;. . . .the corresponding concrete is a name given to things in order to express their possessing that attribute. So when we predicate a concrete name of anything, what really predicate of it is the attribute. [The rest of this paragraph repetitively emphasizes Mill’s view that every attribute is ‘an existence, a sequence, a co-existence, a causation, or a resemblance’.]

It is impossible to imagine any abstract-terms proposition that can’t be transformed into a precisely equivalent proposition with concrete terms—namely,

•the concrete names that connote the attributes themselves, or

•the names of the facts or phenomena on which those attributes are grounded.

To illustrate the latter case, consider a proposition of which only the subject is an abstract name: ‘Thoughtlessness is dangerous’. Thoughtlessness is an attribute grounded on the facts that we call ‘thoughtless actions’; and the proposition is equivalent to ‘Thoughtless actions are dangerous’. Now a couple of propositions in which both terms are abstract names: ‘Whiteness is a colour’, ‘The colour of snow is a whiteness’. These attributes are grounded on sensations, so the equivalent concrete-terms propositions are ‘The sensation of white is one of the items called ‘sensations of colour’ and ‘The visual sensation caused by looking at snow is one of the sensations called sensations of white’. . . . Now two examples in which the concrete terms directly correspond to the abstract names, connoting the attribute that these denote: ‘Prudence is a virtue’ = ‘All prudent persons, in so far as prudent, are virtuous’. ‘Courage is deserving of honour’ = ‘All courageous persons are deserving of honour in so far as they are courageous’, which = ‘All courageous persons deserve an addition to the honour, or a diminution of the disgrace, that would attach to them on other grounds’. [The ‘in so far as’ locution is Mill’s.]

Let’s go into more detail about the ‘prudence’ example. In ‘Prudence is a virtue’ let us replace ‘virtue’ by an equivalent but more definite expression—e.g. ‘a mental quality beneficial to society’ or ‘a mental quality pleasing to God’ or whatever else we use to define ‘virtue’. What the proposition asserts is a •sequence accompanied by •causation, namely that benefit to society (or whatever) is •consequent on prudence and •caused by it. Here is a sequence; but between what? We understand the consequent of the sequence, but what about the antecedent? In connection with prudence, the attribute, two things are to be considered: prudent •persons, who have the attribute, and prudent •conduct, which can be called the foundation of it. [Mill says that neither of these is the attribute, because a prudent person may be a rogue, and behaviour that is prudent may nevertheless have bad consequences that outweigh the prudential good. He continues:] Thus, neither the substance nor the phenomenon—neither the person nor the conduct—is an antecedent which the other term of the sequence always follows. But ‘Prudence is a virtue’ is a universal proposition. What is it, then, that the proposition says is always followed by the effects in question?. . . . ·It’s the relevant mental states and actions·—a correct foresight of consequences, a just estimation of their importance to the object in view, and repression of any thoughtless impulse that would interfere with the deliberate purpose. These are the real antecedent in the •sequence, the real cause in the •causation, asserted by the proposition. They are also the real ground of the attribute of prudence; where these states of mind exist we can predicate prudence without knowing whether any conduct has followed. In this way, every assertion about an attribute can be transformed into an exactly equivalent assertion about the phenomenon that is the ground of the attribute. . . .

NOTES

[1] Bain in commenting on this passage says that the word ‘class’ has two meanings: ‘the class definite, and the class indefinite. The class definite is an enumeration of actual individuals, like the Peers of the Realm, the oceans of the globe, the known planets. . . The class indefinite is not enumerated. Such classes are stars, planets, gold-bearing rocks, men, poets, virtuous. . . In this second meaning of the word, class name and general name are identical. The class name denotes an indefinite number of individuals, and connotes the points of community or likeness.’ The theory I criticise in the text assumes all classes to be definite. I have assumed them to be indefinite, because for the purposes of logic definite classes are almost useless. . . .

[2] He wrote: ‘From hence it may be deduced that the first truths were arbitrarily made by those who first of all imposed names on things. . . . For it is true (for example) that man is a living creature, but only because men chose to impose both those names on the same thing.’

[3] He writes: ‘Men are subject to err not only in affirming and denying, but also in perception and in silent cogitation. . . Tacit errors, or the errors of sense and cogitation, are made by passing from one imagination to the imagination of another different thing; or by feigning that to be past, or future, which never was, nor ever shall be; as when by seeing the image of the sun in water, we imagine the sun itself to be there; or by seeing swords, that there has been, or shall be, fighting, because it used to be so for the most part; or when from promises we imagine the mind of the promiser to be such-and-such. . . . And errors of this sort are common to all things that have sense.’

[4] I fully accept the Law of Relativity, an important truth of which Bain has been in our time the principal expounder and champion; but unlike him I don’t take it to say that we can’t apprehend or be conscious of any fact except by contrasting it with some other positive fact. The needed antithesis. . . .may be between one positive and its negative. . . . The relative opposite of Being, considered as a highest genus, is Nonentity or Nothing; and we do sometimes have reason to consider and discuss things merely in contrast with Nonentity. . . .

Chapter 6. Merely verbal propositions

§1. The proper aim of logic is to lay down how propositions are to be proved, and on my way to that I have had to investigate what they assert that requires proof or can be proved. I examined the conceptualist opinion that a proposition expresses a relation between two ideas; and the extreme nominalist doctrine that it expresses an agreement or disagreement between the meanings of two names. I argued that neither of these is a correct general theory: many propositions are not about names or about ideas. I then examined the different kinds of propositions, and found that apart from those that are merely verbal they assert five different kinds of matters of fact—existence, order in place, order in time, causation, and resemblance. . . .

Notice that I excepted ‘merely verbal’ propositions, which don’t relate to anything properly called a ‘matter of fact’. Since names and their signification are entirely arbitrary, such propositions can’t be (strictly speaking) true or false; they can only be in conformity or disconformity with usage or convention; and the only proof they are capable of is proof of usage—proof that the words have been used by others in the meaning the speaker or writer wants to give them. Yet these propositions have a conspicuous role in philosophy, and their characteristics are as important in logic as those of any of the five other classes of propositions.

If all propositions about the signification of words were as simple and unimportant as. . . .the likes of ‘Cicero is Tully’, there would be little to attract philosophers to attend to them. But the class of merely verbal propositions contains ones that are not like that, including some that at first sight don’t seem to be merely verbal. These latter include a kind of assertions that have been regarded not only as relating to things (·rather than merely to words·) but as being more intimately related to things any other propositions whatever. If you know some philosophy you’ll see that I’m referring to the distinction—much emphasized by the scholastics and still retained under one or another label by most metaphysicians today—between ‘essential’ and ‘accidental’ propositions, and between ‘essential’ and ‘accidental’ properties or attributes.

§2. Almost all metaphysicians before Locke and many since his time have made a great mystery of essential predication and of predicates that are said to be ‘of the essence’ of the subject. Their view went like this:

The essence of a thing is that without which the thing couldn’t exist or be conceived to exist. Thus, rationality is of the essence of man, because man could not be conceived to exist without rationality. The different attributes that make up the essence of a thing x are its essential properties; and a proposition that predicates any of these of x is an essential proposition, which goes deeper into x’s nature, conveying more important information about it, than any other proposition could do. Properties of x that aren’t of its essence are its accidents. They have almost nothing to do with its inmost nature, and a proposition that predicates any of these of x is an accidental proposition.

. . . .The terms ‘essence’ and ‘accident’ were the scholastics’ technical expression of their false views about nature of classification and generalisation. That’s the only possible explanation of their having misunderstood the real nature of the ‘essences’ that loomed so large in their philosophy. They were right in saying that man can’t be conceived without rationality. But though man cannot, a being can be conceived exactly like a man in every way except for rationality and whatever other qualities are tied to it. Thus, the truth of ‘Man cannot be conceived without rationality’—the whole truth—is merely that if a being doesn’t have rationality it doesn’t count as a man. Such a being can be conceived; for all we know, it can exist; but the conventions of language won’t allow it to be called by the name that is reserved for rational beings. Rationality, in short, is involved in the meaning of the word ‘man’; it’s one of the attributes connoted by that name. The ‘essence of man’ simply means all the attributes connoted by ‘man’, and each of those attributes is an essential property of man.

It’s easy for us to see this, but it would have been difficult for persons who thought—as most of the later Aristotelians, including the scholastics, did—that. . . .gold (for instance) was made gold not by

having certain properties that mankind have chosen to attach the word ‘gold’

but by

participating in the nature of a general substance—‘gold in general’—which inheres in every individual piece of gold.

They didn’t think that these universal substances are attached to all general names, but only to some; so their view was that •an object borrows only some of its properties from a universal substance (its essence) and that •the rest belong to it individually (its accidents). The scholastic doctrine of essences lingered on for years after the death of the underlying theory about real ·general· entities corresponding to general terms; and it was left to Locke to convince philosophers that the ‘essences’ of classes were merely the significations of their names. Among Locke’s notable services to philosophy, none was more needed or more valuable than this.

Any proposition ‘Every S is P’, where P stands for an attribute that belongs to the set of attributes connoted by S, is of course true. . . . But it won’t tell you anything you didn’t know already if you know the meaning of S. ‘Every man is a corporeal being’, ‘Every man is a living creature’, ‘Every man is rational’, convey no knowledge to anyone who is already aware of the entire meaning of ‘man’. . . . That every man has the attributes connoted by all these predicates is already asserted when he is called a ‘man’. All the propositions that have been called ‘essential’ are like that; they are in fact identical [see Glossary] propositions.

It is true that a proposition that predicates any attribute, even one implied in the subject-name, is usually understood to imply that there exists a thing corresponding to that name and having the attributes connoted by it; and that implied assertion may convey information even to someone who already understands the meaning of the name. But any information of this sort. . . .is included in the assertion ‘Men exist’. This implication of real existence results from an imperfection of language, namely the ambiguity of the copula ‘is’, which serves (a) as a mark to show that an assertion is being made and (b) as a concrete word connoting existence. So this implication of the actual existence of such a proposition isn’t real. We may say ‘A ghost is a disembodied spirit’ without believing in ghosts. But an accidental or non-essential affirmation does imply the real existence of the subject. . . . ‘A ghost walks the battlements every Tuesday’ has to be understood as implying a belief in ghosts. The meaning of ‘ghost’ implies nothing like this, so the speaker either means nothing or means to assert something that he wants us to believed really happens.

I’ll show later on that when—as in mathematics—any important consequences seem to follow from an ‘essential proposition’, i.e. from a proposition involved in the meaning of a name, what they really flow from is the assumption that the objects so named really exist. Apart from this assumption of real existence, propositions in which the predicate is ‘of the essence’ of the subject. . . .do nothing but unfold the whole or some part of the meaning of the name to those who don’t already know it. So the most useful kind—strictly speaking the only useful kind—of essential propositions are definitions. [Mill adds brief remarks about kinds of definition, and says that definitions ‘will be minutely considered’ in chapter 8.]

§3. According to the view I have presented, no proposition can qualify as ‘essential’ if its subject term is a proper name. Individuals don’t have essences. When the scholastics talked of the ‘essence’ of an individual, they didn’t mean the properties implied in its name, for the names of individuals don’t imply any properties. They counted as ‘of the essence’ of an individual x whatever was of the essence of the species usually assigned to x—i.e. the class that they thought x naturally belonged to. Thus, because ‘Man is a rational being’ is an essential proposition, they said the same thing about ‘Julius Caesar is a rational being’. This followed very naturally if genera and species were to be considered as entities distinct from the individuals composing them—distinct from them and inhering in them. If man was a substance inhering in each individual man, the essence of man (whatever that might mean) was naturally supposed to accompany it; to inhere in John Doe and to form the common essence of Doe and Julius Caesar. That being so, rationality is of the essence of John Doe because it’s of the essence of man. But if man is nothing but the individual men and a name given to them because of certain common properties, what becomes of John Doe’s essence?

A fundamental error is seldom expelled from philosophy by a single victory. It retreats slowly, defends every inch of ground, and after it has been driven from the open country it often retains a footing in some remote safe haven. The essences of individuals were a meaningless illusion caused by a misunderstanding of the essences of classes; yet even Locke, having wiped out the parent error, couldn’t shake himself free from one of its offspring. He distinguished two sorts of essences, real and nominal. His ‘nominal essences’ were the essences of classes, explained nearly as I have just explained them. (Indeed, Book III of Locke’s Essay is a nearly flawless treatise on the connotation of names, except for its view that there are ‘abstract Ideas’, a view that is unfortunately •involved in the wording though not necessarily •connected with the thoughts contained in that immortal Third Book.) But he also admitted ‘real’ essences, i.e. essences of individual objects, which he thought to be the causes of those objects’ sensible properties. We don’t know (he said) what these real essences are (and this acknowledgment made the fiction comparatively harmless); but if we did, we could from them alone demonstrate the sensible properties of the object as the properties of the triangle are demonstrated from the definition of triangle. I’ll return to this theory when I discuss demonstration, and the conditions under which one property of a thing can be demonstrated from another. . . .

§4. So an essential proposition is a purely verbal one. . . .which gives no information except about the subjectname, not the thing. Non-essential or ‘accidental’ propositions can be called real [see Glossary] propositions as opposed to verbal ones. They predicate of a thing some fact that isn’t involved in the signification of the name by which the proposition speaks of it, some attribute not connoted by that name. Propositions of this kind include

•all propositions concerning things individually designated, and

•all general or particular propositions in which the predicate connotes some attribute not connoted by the subject.

If any of these is true, it conveys information that isn’t already involved in the names employed. . . . These are the only propositions that are in themselves instructive, or from which any instructive propositions can be inferred.[1]

[The ‘school logic’ that descended from the scholastics, Mill says, has a widespread reputation for being futile; and the biggest single source of this is probably the practice of discussing predication and inference almost entirely with examples that are essential propositions. Animals are bodies, and bodies are substances, therefore animals are substances—what triviality! He concludes:] I have, therefore in this work avoided using essential propositions as examples, except where there is a special point in using them.

§5. As regards propositions that do convey information,. . . .there are two ways of looking at these, or rather at the general propositions among them: we can look at them as portions of •theoretical truth or as memoranda for •practical use. Depending on which of these we adopt, their import can be conveniently expressed in one or other of two formulas.

(i) The formula I have been using is best adapted to express the import of the proposition as an item of theoretical knowledge: ‘All men are mortal’ means that the attributes of man are always accompanied by the attribute mortality; ‘No men are gods’ means that the attributes of man are never accompanied by all the attributes signified by ‘god’. (ii) But when a proposition is considered as a memorandum for practical use, a different way of expressing the same meaning is better adapted to indicate the job the proposition is doing. The practical use of a proposition is to inform or remind us of what we have to expect in any individual case that falls within the scope of the proposition. In that context, ‘All men are mortal’ means that the attributes of man are evidence of—are a mark of—mortality; an indication by which the presence of mortality is made manifest. ‘No men are gods’ means that the attributes of man are a mark or evidence that at least some of the attributes understood to belong to a god are not there; that where the former are, we shouldn’t expect to find the latter.

These two forms of expression are basically equivalent; but (i) points our attention more directly to what a proposition means, (ii) to how the proposition is to be used.

Now reasoning is a process into which propositions enter not as ultimate results but as means to establishing other propositions. So we can expect that the role of propositions in reasoning is best expressed by (ii) the way of exhibiting a general proposition’s import that brings out its application to practical use. In the theory of reasoning, therefore, we pretty well have to think of a proposition as saying that one fact or phenomenon is a mark or evidence of another fact or phenomenon. For the purposes of that theory, the best way to define a proposition’s import is not the one that most clearly shows what the proposition is in itself, but the one that distinctly suggests how it can be made available for advancing from it to other propositions.

NOTES

[1] This distinction corresponds to the one drawn by Kant and other metaphysicians between what they call ‘analytic’ and ‘synthetic’ judgments, the former being the ones that can be derived from the meanings of the terms used.

Chapter 7. The nature of classification. The five predicables

§1. In discussing the nature of general propositions I have made much less use than logicians usually do of the ideas of a class and of classification—ideas that have, since the realist doctrine of general substances went out of vogue, been the basis for almost every attempt at a philosophical theory of general terms and general propositions. I have considered general names as having a meaning, quite independently of their being the names of classes. Whether a general name names a class is utterly irrelevant to its signification; it makes no difference whether there are many objects, or only one, or none, to which it happens to be applicable. The word ‘god’ is as much a general term to the Christian or Jew as to the polytheist; and ‘dragon’, ‘hippogriff’, ‘chimera’, ‘mermaid’, ‘ghost’ are general names, just as much so as if real objects existed corresponding to them. If a name’s signification is constituted by attributes, it is potentially a name of indefinitely many objects; but it needn’t actually be the name of any; and if of any, it may be the name of only one. As soon as we use a name to connote attributes, the things (if any) that have those attributes are thereby constituted a class. But in predicating the name we predicate only the attributes; and in many cases the fact of belonging to a class doesn’t make an appearance.

However, although predication doesn’t presuppose classification, and although the theory of names and propositions is only cluttered by intruding the idea of classification into it, classification is nevertheless closely connected with the use of general names. By every general name that we introduce, we create a class if there are any things, real or imaginary, to compose it, i.e. any things corresponding to the signification of the name. Thus, classes mostly owe their existence to general language. But general language sometimes, though not very often, owes its existence to classes. A general name is mostly introduced

because we have a signification to express by it; i.e. because we need a word by which to predicate the attributes that it connotes.

But sometimes a general name is introduced

because we have found it convenient to create a class; i.e. because we have thought it useful in our thinking that a certain group of objects should be thought of together.

For example, a naturalist has scientific reasons to sort the animal or vegetable creation into certain groups rather than into any others, and he needs a name to bind each of his groups together, so to speak. But don’t think that names introduced in this way are in any way different from other connotative names in how they signify. The classes they denote are, as much as any other classes, constituted by certain common attributes, and their names signify those attributes and nothing else. . . .

The principles that ought to regulate classification, as a logical process subservient to the investigation of truth, can’t be usefully discussed until much later in my inquiry. But I have to deal here with classification considered as resulting from (and implied in) the fact of employing general language; if I didn’t, I would be leaving the theory of general names and of their use in predication in a mutilated and formless state.

§2. This part of the theory of general language is the subject of what is called the doctrine of the predicables, a set of distinctions handed down from Aristotle and his follower Porphyry, many of which have taken root in scientific terminology and some even in ordinary informal language. The predicables are a five-fold division of general names, base not on •differences in their meanings, i.e. on the attributes they connote, but on •differences in the kind of class they denote. There are five different sorts of class-name that we can use in predication:

a genus of the thing

a species

a differentia

a proprium

an accidens

[Those labels are Latin; Mill also gives the Greek for each.] Notice that when you put one of these labels on an attribute what you’re talking about is not •what the attribute is like in itself, but •what relation it has to the subject you are predicating it on. . . . ‘Animal’, for instance, is a genus with respect to man or John, a species with respect to substance or being. ‘Rectangular’ is one of the differentiae of a geometrical square; it is merely an accidens of the table at which I am writing. The words ‘genus’, ‘species’, etc. are therefore relative terms: they are names applied to certain predicates to express the relation between them and some given subject. And we’ll see that the relation is grounded not on what the predicate connotes but on •the class it denotes and •the place that class has in some given classification relatively to the particular subject.

§3. Two of these names, ‘genus’ and ‘species’, are used by naturalists in technical senses not precisely agreeing with their philosophical meaning, and have also acquired popular [see Glossary] meanings that are much more general still. In this popular sense, any two classes can be called ‘genus’ and ‘species’ if one includes the other and more. For instance, animal and man; man and mathematician. . . . Taste is a genus of which sweet taste, sour taste, salt taste etc. are species. Virtue is a genus; justice, prudence, courage, fortitude, generosity etc. are its species.

A class that is a genus relative to the sub-classes or species included in it can itself be a species relative to a more comprehensive genus. [Examples are then given.]

. . . .The Aristotelian logicians used ‘genus’ and ‘species’ in a more restricted sense. They didn’t count every case of class/subclass as one of genus/species; for example, they wouldn’t say that man is a species belonging to a genus biped. . . . They held that genus and species must be ‘of the essence’ of the subject, and they expressed this also by saying that genus must differ from species in kind. Animal was of the essence of man; biped was not. And in every classification they considered some one class as the lowest species—man was a lowest species, for instance. Any further divisions of the class—e.g. man into white, black, and red man—they didn’t count as species.

I showed in chapter 6 that the distinction between •the essence of a class and •the attributes that aren’t of its essence. . . .is in fact merely the difference between •attributes of the class that are involved in the signification of the class-name and •attributes that aren’t. And I showed that as applied to individuals the word ‘essence’ has no meaning except in the context of the exploded tenets of the realists. . . .

Is this merely verbal difference all there is to the line the scholastics drew between the class-pairs that they would count as genus/species and the ones that they wouldn’t? Is it an error to regard some of the differences that exist among objects as differences in kind (in genus or species), and others only as differences in the accidents? Were the scholastics right or wrong in calling some classes ‘kinds’ and others not? We’ll see that the Aristotelians did mean something by this distinction—something important—but they didn’t have it clear in their minds, which is why they expressed it in terms of ‘essences’ and other unsatisfactory language.

§4. It’s a fundamental principle in logic that the power of framing [see Glossary] classes is unlimited, as long as there’s some difference, however tiny, to base a distinction on. Take any attribute A whatever: and if some things have it and others don’t, we can take it as the basis for dividing all things into two classes; and we actually do this the moment we create a name that connotes the attribute. So the number of possible classes is boundless; and there are as many actual classes (of real or of imaginary things) as there are general names (including the negative ones).

[The labels ‘K’ and ‘NK’ are reminders—not by Mill—of ‘difference in kind’ and ‘difference not in kind’.] But if we contemplate any one of the classes so formed, such as the classes animal or plant, sulphur or phosphorus, white or red, and ask ·what I’ll call ‘the Question’·:

What features are possessed by every individual in this class and by no individual outside it?

we’ll get two very different sorts of answer, depending on what class is in question. For NK classes, the things in the class differ from other things only in certain particulars that we could list; whereas for K classes there are more differences than we could list, more even than we have much chance of knowing. Ask the Question about white things, for example: they have nothing in common except their whiteness (and perhaps some features connected somehow with whiteness). But now ask it about some K class—plants or phosphorus: many centuries of research haven’t given us a complete list of the common properties of animals or of portions of phosphorus. And we don’t assume that we can complete the list: we go on making new observations and experiments, in confidence that we’ll discover new properties that weren’t implied in any of the ones we previously knew. So far as we’re concerned, the list of all the common properties might as well be infinite. Contrast that with a research proposal to investigate the common properties of all things in an NK class, e.g. having same colour, or the same shape, or the same specific gravity! The absurdity would be obvious. We have no reason to think that any such common properties exist, apart from ones that are •involved in the defining attribute or •derivable from it by some law of causation. . . . [The phrase ‘involved in the defining attribute’ replaces Mill’s puzzling ‘involved in the supposition itself’.]

There’s nothing wrong with saying that a K classification answers to a much more radical distinction in the things themselves than does an NK classification. And if you want to say that K classifications are made •by nature while NK ones are made •by us for our convenience, you’ll be right, provided you mean only this:

•(K) Where a certain apparent difference between things answers to who-knows-how-many other differences,. . . .some involving properties not yet discovered, it is not optional but imperative to recognise this difference as the basis for a specific distinction; whereas

•(NK) differences that are merely finite and determinate, like those designated by the words ‘white’, ‘black’, or ‘red’, can be disregarded if the purpose for which the classification is made doesn’t require that they be attended to.

The differences are made by nature in both cases; the recognition of them as bases for classification and naming is done by man in both cases. Where they differ is in this: in one case (K) the purposes of language and of classification would be subverted if no notice were taken of the differences ·that define the class·, while in the other case (NK) the need to take notice of the defining difference depends on how important the relevant qualities are to us.

Now, these (K) classes, distinguished by unknown multitudes of properties, and not solely (NK) by a few determinate ones—which (K) are separated off from one another by an unfathomable chasm instead of (NK) by a mere ordinary ditch with a visible bottom—are the only classes that the Aristotelian logicians counted as genera or species. . . . They were justified in drawing a broad line of separation between these two kinds of classes and of class-distinctions; and I shall not only •retain the division itself but also •continue to express it in their language. According to that language, the ‘proximate’ (or lowest) kind to which any individual can be assigned is called its species: so Isaac Newton would be said to be of the species man. He also belongs to many sub-classes included in the class man—e.g. Christian, Englishman, Mathematician. But though these are distinct classes they are not, in our sense of the word, distinct ‘kinds’ of men. A Christian differs from other human beings; but he differs only in the attribute that the word ‘Christian’ expresses—belief in Christianity and whatever else that implies. . . . We would never think of asking ‘What properties, unconnected with Christianity either as cause or effect, are possessed by all Christians and by no-one else?’; whereas physiologists are perpetually asking the analogous question about men, and aren’t likely ever to have a complete answer. So man is a species and Christian is not.

I’m not denying that there may be different kinds— different ‘species’ in the Aristotelian sense—of men. The various races and temperaments, the two sexes, and even the various ages, may be differences of ‘kind’ in my meaning of the term. I don’t say that they are so. The progress of physiology suggests that they are not, making it look likely that the real differences between different races, sexes, etc. are (NK) causally natural consequences of a small number of primary differences that can be precisely determined and that account for all the rest. If that is so, these are not (K) distinctions in kind, any more than are the differences of Christian, Jew, Moslem, and pagan, which also carry many consequences along with them. That is how classes are often mistaken for real kinds and then found not to be so. But if it turned out that the differences couldn’t be thus accounted for, then Caucasian, Mongolian, Negro, etc. really would be different kinds of human beings, and entitled to be ranked as species by the logician. But not by the naturalist, because (I repeat) the word ‘species’ is used with different significations in logic and in natural history. Naturalists usually don’t put two organisms in different species if they are thought to have descended from the same stock. But that is an artificial sense given to the word for the technical purposes of that one science. To the logician, if a Negro and a white man differ in the same way (even if not to the same extent) as a horse differs from a camel—i.e. if their differences are inexhaustible and not traceable to any common cause—they are different species, whether or not they have ancestors in common. But if their differences can all be traced to climate and habits, or to some one or a few special differences in structure, logician doesn’t assign them to different species. [This work first appeared 16 years before Darwin’s The Origin of Species. Three or four editions of System post-dated Origin, but weren’t altered in the light of it.]

When the lowest species—the ‘proximate kind’—to which an individual belongs has been ascertained, the properties common to that kind must include all the common properties of every other real kind to which the individual can be assigned. Let the individual be Socrates, and the proximate kind man. Animal is also a real kind, and includes Socrates; and since all men are animals the properties common to animals are some of the common properties of the sub-class man. And any class that includes Socrates without including man is (NK) not a real kind—e.g. the class of flat-nosed animals. If it turned out that

•all flat-nosed animals have common properties that aren’t implied in their flat noses and aren’t possessed by all animals whatever; so that

•a flat nose was a marker for indefinitely many other special features not deducible from flat-nosedness by an ascertainable law,

then out of the class man we could cut (K) another class, flat-nosed man, which would be a kind. In that case, though, man would not be the proximate kind after all. . . .

Now we can fix the logical meaning of the terms ‘genus’ and ‘species’. Every class that is (K) a real kind—i.e. is distinguished from all other classes by indeterminately many properties not derivable from one another—is either a genus or a species. A kind that isn’t divisible into other kinds is a species, and can’t be a genus because it has no species under it. . . . But every kind that can be divided into real kinds (as animal into mammal, bird, fish, etc., or bird into various species of birds) is •a genus to all below it and •a species to all genera in which it is included. With that settled, we can move on to the three remaining predicables, differentia, proprium, and accidens.

§5. The word ‘differentia’ is tied to ‘genus’ and ‘species’. Everyone agrees that a differentia is an attribute that distinguishes a given species from every other species of the same genus. So far so good, but which of the distinguishing attributes is the differentia? ·The question arises· because every kind (and thus every species) is distinguished from other kinds by indefinitely many attributes. Man is a species of the genus animal, and logicians usually assign rational as the differentia (or rationality; it doesn’t matter here whether we use the concrete or the abstract form of the word); man is also a cooking animal, and the only one. Since this is another attribute marking off man from other species of the same genus, would it serve equally well as a differentia? The Aristotelians say No, because they held that the differentia must be ‘of the essence of’ the subject, just as the genus and species are.

We saw that when a genus and species are said to be ‘of the essence’ of the things they contain, there’s a vestige of a meaning based on the nature of the things themselves ·and not merely the words used to express them·. When the scholastics distinguished things’ ‘essences’ from their ‘accidents’, they were confusedly distinguishing (K) differences of kind from (NK) differences that are not of kind; they meant to say that genera and species must (K) be kinds. Their notion of ‘the essence of’ a thing x was a vague notion of a something that

•makes x be what it is, i.e.

•makes it the kind of thing that it is, i.e.

•causes it to have all the variety of properties that distinguish its kind.

But when this was looked into more closely, nobody could discover •what caused the thing to have all those properties, or even •that anything causes it to have them. Logicians, however, didn’t like to admit this, and being unable to detect what makes the thing be what it is, settled for knowing what makes it be what it is called. Of the innumerable properties, many of them unknown, that are common to the class man, only a few are connoted by its name; these few will naturally have been distinguished from the rest because they are more obvious or more important. And these are the properties that logicians seized on and called ‘the essence of’ the species; and they went even further in the case of a lowest species and called them ‘the essence of’ the individual too, because they held that the species contained the whole essence of the thing. Metaphysics, that fertile field of delusion propagated by language, doesn’t offer a more notable example of such delusion than this one. . . .

So the distinction between differentia, proprium, and accidens is grounded not in the nature of things but in the connotation of names. If we want to know what it is, that’s where we must look.

From the fact that the genus includes the species—i.e. denotes more than the species. . . .—it follows that the species must connote more than the genus. It must connote all the attributes the genus connotes, or there would be nothing to prevent it from denoting individuals not included in the genus. And it must connote something besides what the genus connotes, because otherwise it would include the whole genus. ‘Animal’ denotes all the individuals denoted by ‘man’ and many more. So ‘man’ must connote all that ‘animal’ connotes, or there could be men who weren’t animals. And it must connote something more than ‘animal’ connotes, or all animals would be men. This surplus of connotation—what the species connotes over and above the connotation of the genus—is the differentia; or, in other words, the differentia is what must be added to the connotation of the genus to complete the connotation of the species.

The word ‘man’, for instance, in addition to what it connotes in common with ‘animal’, also connotes rationality, and some approximation to the external shape that we all know but don’t have any name for except ‘human shape’. So the differentia of man in relation to the genus animal is •that outward shape and •the possession of reason. The Aristotelians said it was the possession of reason alone, without the outward shape. But if they kept to this, they’d have been obliged to call the Houyhnhnms men. [They are virtuous horses in one of Swift’s Gulliver tales]. The question never arose, and they didn’t have to decide how such a case would have affected their notion of essentiality. . . .

§6. Let’s not give the notion of differentia too narrow a range. It can happen that a species is marked off within a genus by more than one differentia’, and which one is selected depends on the purposes and methods of the person making the classification. A naturalist surveys the various kinds of animals looking for the classification of them that fits best with the order in which he thinks it desirable for us to think of them. With that aim, he favours a system in which there’s a basic division between warm-blooded and cold-blooded animals; or between animals that breathe with lungs and those that breathe with gills;. . . .or between those that walk flat-footed and those that walk on their toes. . . . In doing this, he creates new classes that aren’t the ones animals are assigned to in casual conversation; no-one would think of classifying animals in the naturalist’s terms unless we were ruled by the need for scientific convenience. . . .

Practical convenience justifies us in making the main lines of our classification not coincide with any distinction of kind, thus creating ‘genera’ and ‘species’ in the popular sense that aren’t ‘genera’ or ‘species’ in the rigorous sense. So we must also be justified, when our genera and species are real genera and species, in marking the distinction between species and genus in whatever suits our practical convenience. If we cut the species man out of the genus animal, for instance, intending the cut to be governed by rationality, then ·for us· rationality is the differentia of the species man. But if as naturalists we find it scientifically convenient to mark out man from the genus animal by ‘four incisors in each jaw, tusks solitary, and erect posture’, then ‘man’ as used by us as naturalists connotes not rationality but those three other properties. . . . The upshot of all this is as follows:

In any genus/species pair, the name of the species must •be connotative and must •connote the differentia; but the connotation can be special—not involved in the ordinary use of the word but given to it when it is used as a term of art or science.

[Mill remarks that if for a given genus/species pair two different differentiae have some currency, we might discover something that fits one differentia and not the other—e.g. a rational animal that doesn’t fit the differentia used by the naturalist Linnaeus. Would it be a man?]

Words that aren’t otherwise connotative can acquire a special or technical connotation in the way I have described. ‘Whiteness’ (I repeat) doesn’t connote anything; it merely denotes the attribute corresponding to a certain sensation; but if you’re writing an article about the classification of colours, and want to underline the particular place of whiteness in your scheme, you can define it as ‘the colour produced by the mixture of all the simple rays’; and this fact, though not implied in the ordinary meaning of ‘whiteness’, is part of its meaning in your article, where it is the differentia of the species.

So the differentia of a species can be defined as: the part of the connotation of the species-name, whether ordinary or special and technical, that distinguishes the species from all other species of the genus to which we are assigning it.

§7. Having dealt with genus, species, and differentia, we won’t find it hard to get clear conception of how proprium differs from accidens, and how those two differ from the first three.

In the Aristotelian terminology genus and differentia are ‘of the essence’ of the subject, which really means (as we saw) that the properties signified by the genus and those signified by the differentia are part of the connotation of the name denoting the species. Proprium and accidens, on the other hand, are not part of the essence, and are predicated of the species only accidentally [see Glossary]. Both are ‘accidents’ in the broader sense in which a thing’s accidents are opposed to its essence; but in the ·Aristotelian· doctrine of the Predicables ‘accidens’ is used only for one sort of accident, ‘proprium’ for another. Proprium, the scholastics said, is predicated accidentally but necessarily, or—as they further explain it—it signifies an attribute that flows from the essence without being a part of it, so that it is inseparably attached to the species. For example, all the properties of a triangle that aren’t part of its definition must be possessed by anything that fits that definition. Accidens, on the other hand, has no connection with the essence; it can come and go without affecting the species. . . . Whether or not we actually encounter a member of the species that doesn’t have the accidens, we can think of such a thing without having to think of it as no longer belonging to that species.

So we can define the proprium of a species as: any attribute that isn’t connoted by the species-name but follows from some attribute that the species-name does connote. This applies whether the classification in question is an ordinary everyday one or a special one adopted for a special purpose.

There are two kinds of proprium, because there are two ways for one attribute to ‘follow from’ another: it may follow •as a conclusion follows premises or •as an effect follows a cause. For example, the attribute of having the opposite sides equal isn’t connoted by the word ‘parallelogram’ but follows from ones that are connoted by it;. . . .so that attribute is a proprium1 of the class parallelogram, by which I mean that it follows demonstratively from the connoted attributes. The attribute of being able to understand language is a proprium of the species man, because it follows from an attribute that ‘man’ does connote, namely rationality. But this is a proprium2, by which I mean that it follows causally from rationality. Many questions arise, which I’ll deal with in Books II and III. All I need to say here is that whether a proprium follows by •demonstration or by •causation it follows necessarily—i.e. by some law that is a part of the constitution either of •our thinking faculty or of •the universe.

§8. The fifth and last predicable, accidens, includes all the attributes of a thing that aren’t involved in the signification of its species name (whether ordinary or technical), and aren’t known to have any necessary connection with attributes that are involved in it. They are commonly divided into two groups. Inseparable accidents. . . .are ones that aren’t necessary to the species though they are in fact possessed by all the species’ members. Blackness is an attribute of a crow, and (as far as we know) of all crows. But if we came across a race of white birds that were like crows in every other way, we would say ‘These are white crows’—not ‘These are not crows’. . . . We conceive a white crow, and don’t know of any reason why such an animal should not exist, but we don’t know of any real white crows; so in our present state of knowledge blackness counts as an accident, but an inseparable accident, of the species crow.

Separable accidents are attributes that aren’t possessed by all the members of the species all the time: as well as not being necessary ·to the species· they aren’t even universal ·throughout it·. . . . Thus the colour of a European is a separable accident of the species man, because it isn’t an attribute of all human creatures. Being born is also a separable accident of the species man: it’s an attribute that all human beings possess, but only at one particular time for each. . . .

Chapter 8. Definition

§1. One part of the theory of names and propositions remains to be discussed: the theory of definitions. When I discussed what I called ‘purely verbal’ propositions in chapter 6, I said a little about definitions, which are the most important purely verbal propositions; but there would have been no point in trying to deal with them thoroughly until I had dealt with classification.

The simplest and best notion of definition is: a proposition that declares the meaning of a word—it may be the ordinary popular meaning or a special meaning that the speaker or writer wants to give to the word for his own purposes.

Obviously, words that have no meaning can’t be defined. So proper names can’t be defined. You can’t define ‘John Doe’ by saying ‘John Doe is the son of General Doe’ or ‘That’s John Doe over there, crossing the street’. These propositions may make known who is the particular man to whom the name belongs, but you can do that even more unambiguously by pointing to him!

The meaning of a connotative name, as I have already said several times, is its connotation; and the definition of a connotative name is the proposition that says what that connotation is. It can do this either directly or indirectly. The word ‘man’ is defined directly by a proposition of the form: ‘“Man” is a name connoting such and such attributes’, or ‘. . . is a name which, when predicated of a thing, signifies its possession of such and such attributes’, or ‘Man is everything that has such and such attributes—e.g. that has corporeity, organisation, life, rationality, and certain special features of external shape’.

This is the most precise and least ambiguous type of definition; but it’s long-winded, and also too technical for common discourse. The more usual indirect way of declaring the connotation of a name is to predicate of it another name or names of known signification, which connote the same collection of attributes. This can be done either by using another exactly synonymous connotative name—e.g. ‘Man is a human being’, which isn’t commonly counted as a definition at all; or by using two or more connotative names that jointly make up the whole connotation of the name being defined. In this last case, we can use as many connotative names as there are attributes, one for each, as in

•‘Man is a corporeal, organized, animated, rational being, shaped so and so’;

or we can shorten the definition by using names that connote several of the attributes at once, as in

•‘Man is a rational animal, shaped so and so’.

The definition of a name, on this view of it, is the sum total of all the essential propositions that can be formulated with that name as subject. All the propositions whose truth is implied in the name, all that we are made aware of by merely hearing the name, are included in the definition if it is complete; and we can extract them from it without the aid of any other premises. . . . So Condillac and other writers had reason to assert that a definition is an analysis. Breaking a complex down into the elements of which it is compounded—that’s what ‘analysis’ means; and it’s also what we do when we ·define a word, i.e.· replace one word that connotes a set of attributes collectively by two or more that connote them singly or in smaller groups.

§2. How, then, are we to define a name that connotes only a single attribute? For example, ‘white’, which connotes nothing but whiteness; ‘rational’, which connotes nothing but the possession of reason. You might think that the meaning of such names could be declared in only two ways: •by a synonymous term, if one be found; or •in the direct way I have described, e.g. ‘“White” is a name connoting the attribute whiteness.’ ·Before jumping to that conclusion·, however, let us see whether we can go further with breaking down the word’s meaning into several parts. Setting aside the question of whether we can do this with ‘white’, it’s obvious that we can do more to explain the meaning of ‘rational’ than merely to say ‘Rational is that which possesses the attribute of reason’, because the attribute of reason can itself be defined. This brings us to the topic of the definitions of attributes, or rather of the names of attributes, i.e. of abstract names.

Two kinds of case are unproblematic. (a) If N is an attribute-name that is itself connotative, i.e. expresses attributes of the named attribute, then like other connotative names it is defined by declaring its connotation. Thus ‘fault’ may be defined as ‘a quality productive of evil or inconvenience’. (b) Sometimes the attribute ·whose name is· to be defined is not one attribute but a union of several; and in that case we need only to put together the names of all those attributes taken separately, and that will be a definition of the name that belongs to them all taken together—a definition that corresponds exactly to the definition of the corresponding concrete name. . . . For example, if the definition of ‘human being’ is ‘a being that is corporeal, animated, rational, shaped so and so’, then the definition of ‘humanity’ will be ‘corporeity and animal life, combined with rationality, and with such and such a shape’.

What about the case where the abstract name does not express an aggregation of attributes, but only a single attribute? Here we must remember that every attribute is grounded on some fact or phenomenon which is the sole source of its meaning—what here I called the ‘foundation’ of the attribute, which we must now look to for its definition. The foundation of an attribute can be a phenomenon of any degree of complexity, consisting of many different parts, either co-existent or in succession; and we can define ·the name of· the attribute by analysing the phenomenon into these parts. ‘Eloquence’, for example, is the name of a single attribute, but this attribute is grounded on complicated external •effects, flowing from •acts of the person whom we credit with eloquence; and by resolving this phenomenon into its two parts, the •cause and the •effect, we get a definition of ‘eloquence’, namely ‘the power of influencing the feelings by speech or writing’.

. . . .Suppose, though, that the fact on which an attribute is grounded is one of our simple feelings or states of consciousness, and therefore can’t be analysed. Even then, the names both of the attribute and of object that has it can be defined—or rather they could be defined if all our simple feelings had names. ‘Whiteness’ can be defined as ‘the property or power of exciting the sensation of white’. A ‘white object’ may be defined as ‘an object that arouses the sensation of white’. The only names that can’t be defined because their meaning can’t be analysed are the names of the simple feelings themselves. These are in the same position as proper names. They aren’t unmeaning as proper names are, for the words ‘sensation of white’ signify that the sensation I’m applying this to resembles other sensations that remember having experienced and called by that name. But as we have no words by which to recall those former sensations—except the very word we’re trying to define, or some other that requires definition just as much—words cannot unfold the signification of this class of names; and we have to make a direct appeal to the personal experience of the person we are speaking to.

§3. Having stated what I think is the true idea of a definition, I shall now examine some opinions of philosophers, and some popular conceptions, that conflict with that idea.

I have contended that the only adequate definition of a name is one that declares all the facts that the name involves in its signification. But for most people a definition doesn’t aim to do so much; all they want from a definition is a guide to the correct use of the defined word—a protection against applying it in a manner inconsistent with custom and convention. So they’ll accept as a sufficient definition of a term anything that accurately points to what the term denotes, even if it doesn’t take in the whole—perhaps doesn’t even take in any part—of what it connotes. This gives rise to two sorts of imperfect or unscientific definition: (i) essential but incomplete definitions, in which a connotative name is defined by a part of its connotation; and (ii) accidental definitions or descriptions in which the name is ‘defined’ by something that isn’t part of the connotation at all.

(1) Here’s an example of the first kind of imperfect definition: ‘Man is a rational animal’. We can’t regard this a complete definition of ‘man’, because it would require us to call the Houyhnhnms ‘men’; but as there happen to be no Houyhnhnms, this imperfect definition satisfactorily marks the objects that are actually denoted by ‘man’. . . . Such definitions, however, are always liable to be overthrown by the discovery of new objects in nature. . . .

§4. Although that first kind of imperfect definition. . . .has been considered by the ancients and by logicians in general as a complete definition, they have always insisted that the attributes employed should be part of the connotation. The rule was that the definition must be drawn from the essence of the class, not including attributes not connoted by the class name. The second kind of imperfect definition, therefore, in which the name of a class is defined by some of its accidents—i.e. by attributes that aren’t included in its connotation—has been counted by logicians not as a genuine definition but as a ‘description’.

(2) This second kind of imperfect definition, however, has the same sources as the first, namely, a willingness to accept as a definition anything that enables us •to pick out the things denoted by the name and thus •to use the term in predication without deviating from accepted usage. This end is achieved by stating any combination of attributes that is possessed by everything in the class and nothing outside it, even if separate attributes within that combination are shared with some things outside the class. All that is needed is for the definition (or description) to be exactly co-extensive [see Glossary] with the name it professes to define; and that can be achieved even if the specified attributes have no connection with the ones that mankind had in view when they formed or recognised the class and gave it a name. By this test, the following are correct definitions of ‘man’: ‘Man is a mammiferous animal’ [= ‘animal that has mammary glands’], ‘. . . an animal that has two hands’, ‘. . . an animal that cooks its food’, ‘. . . a featherless biped’.

What would otherwise be a mere description can be raised to the rank of a real definition by the special purpose that the speaker or writer has in view. . . . This has actually happened with one of examples I have given: ‘Man is a mammiferous animal with two hands’ is the scientific definition of ‘man’ in Georges Cuvier’s classification of the animal kingdom.

Although such a definition does declare the meaning that the writer or speaker is giving to the word, that wasn’t his purpose in offering the definition. What he was aiming at was to present not a •name but a •classification. The special meaning that Cuvier assigned to ’man’. . . .was a by-product of his plan of arranging animals into classes according to a certain set of distinctions. The definition of ‘man’ according to its ordinary connotation. . . .wouldn’t have assigned the species its place in that particular classification. . . .

Scientific definitions—whether of scientific terms or of common words used in a scientific sense—are mostly of the kind I have been discussing: their main purpose is to serve as landmarks in scientific classification. And as scientific knowledge advances the classifications in any science are continually modified, so the scientific definitions are also constantly varying. The word ‘acid’ is a striking example of this. As experimental discovery advanced, the substances classed with acids have been constantly multiplying, so that the attributes connoted by ‘acid’ have become fewer. At first it connoted the attributes:

(i) combining with an alkali to form a neutral substance (called a salt);

(ii) being compounded of a base and oxygen;

(iii) being caustic to the taste and touch;

(iv) being fluid.

The true analysis of muriatic acid into chlorine and hydrogen caused (ii) to be excluded from the connotation. The same discovery fixed the attention of chemists on hydrogen as an important element in acids; more recent discoveries have revealed its presence in sulphuric, nitric, and many other acids where it hadn’t been suspected to be; so that there’s now a tendency to include the presence of hydrogen in the connotation of ‘acid’. . . . (iii) Causticity and (iv) fluidity have long since been excluded from the characteristics of the class, by the inclusion of silica and many other substances in it. So that now (i) forming neutral bodies by combination with alkalis, together with electro-chemical features that this is supposed to imply, are the only differentiae that give the word ‘acid’ its fixed connotation as a term of chemical science. . . .

In the same way that a technical definition aims to expound the artificial classification out of which it grows, the Aristotelian logicians seem to have imagined that an ordinary definition aims to expound the ordinary natural classification of things, namely the division of them into kinds, and to show the place that each kind has among other kinds—which ones it is superior to, inferior to, or collateral with. This notion would account for the rule that all definition must be by genus and differentia, and would also explain why a single differentia was deemed sufficient. But I have shown that a distinction of kind can’t be expounded or expressed in words: calling something a ‘kind’ means that the properties that distinguish it don’t grow out of one another, and therefore can’t be expressed in words, even by implication, except by enumerating them all. And we don’t know them all, and probably never will; so it’s idle to look to this as one of the purposes of a definition. Whereas if the definition of a kind has only to indicate what kinds include it or are included in it, any definition that expounds the connotation of the name will do this. . . .

[At this point Mill has a very long footnote discussing Bain’s view that a proper definition should list all the ultimate properties of the things the definition is to fit. Bain writes:

‘When we are told that diamond, which we know to be a transparent, glittering, hard, and high-priced substance, is composed of carbon, and is combustible, we must put these additional properties on the same level as the rest; to us they are henceforth connoted by the name.’

Mill remarks that this gives to ‘Diamonds are composed of carbon’ the status of a merely verbal proposition. More generally, Mill objects to saying that the meaning of a word includes items that are known only to a few specialists.]

§5. . . . .I turn now to an ancient doctrine, once generally prevalent and still alive, which I regard as the source of much of the obscurity hanging over some of the most important processes of the understanding in the pursuit of truth. According to this doctrine, there are two sorts of definitions: definitions of •names and definitions of •things. The former are intended to explain the meaning of a term; the latter to explain the nature of a thing—and these are incomparably more important than the others.

This was the view of the ancient philosophers and of their followers except for the nominalists; but the spirit of modern metaphysics has until recently been on the whole a nominalist spirit, so that the notion of definitions of things has retreated somewhat. It still continues to breed confusion in logic—through its consequences rather than through itself—and the doctrine itself now and then breaks out, and has appeared, surprisingly, in Whately’s justly admired Logic. I reviewed that work in the Westminster Review for January 1828 [when he was 21 years old]; I don’t now agree with all of that review, but I stand by the following passage from it [just the next paragraph]:

The distinction between nominal and real definitions, between definitions of words and what are called definitions of things—though it fits the ideas of most of the Aristotelian logicians—can’t be maintained, it seems to me. I don’t think that any definition is ever intended to ‘explain and unfold the nature of a thing’. Of the writers who have thought there are definitions of things, none have discovered any criterion to distinguish •the definition of a thing from •any other proposition about the thing. They say ‘The definition unfolds the nature of the thing’; but no definition can unfold its whole nature, and every proposition in which something is predicated of the thing unfolds part of its nature. [Mill then makes the point that a single sentence might convey both •the claim that there exist things of a certain kind and •a declaration of the meaning of the kind’s name. He continues:] But it’s not correct to call such a sentence a special kind of definition. Really it’s a definition and something more. . . .

So there’s a real distinction between definitions of names and so-called ‘definitions of things’; but it’s just that the latter covertly assert a matter of fact along with the meaning of a name. This covert assertion is not a definition, but a postulate. The definition is a mere identical [see Glossary] proposition, which gives information only about the use of language, and from which no matters of fact can possibly be inferred. The accompanying postulate does affirm a fact that might be anything from trivial to vastly important. It affirms the actual or possible existence of things having the attributes set forth in the definition; and this, if true, may be the foundation for a whole structure of scientific truth.

As I keep saying, the philosophers who overthrew realism didn’t get rid of all its consequences; they retained in their own philosophy many propositions that could have a rational meaning only in the context of a realist system. It had been handed down from Aristotle and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This did well enough as long as a definition was considered to be a proposition ‘unfolding the nature of the thing’. But then came Hobbes, who rejected utterly the notion that a definition. . . .does anything but state the meaning of a name; yet he continued to affirm as sweepingly as any of his predecessors that the basic premises of mathematics, and even of all science, are definitions. This implies a strange paradox: systems of scientific truth— indeed, all truths that we reach by reasoning—are deduced from the arbitrary conventions of mankind concerning the signification of words!

To make credible the doctrine that definitions are the premises of scientific knowledge, the proviso is sometimes added that this holds only if the definitions are made in a way that fits the phenomena of nature—i.e. only if they give words meanings that suit objects actually existing. But this is just another example of retaining the words of a refuted doctrine by radically changing their meaning. We’re being told that you can infer physical facts from •the meaning of a name, provided the name has corresponding to it •an existing thing. But in that case, what is the real premise of the inference—the existence of a thing having the properties, or the existence of a name meaning them?

Consider Euclid’s definition of circle. This consists of two propositions:

(a) ‘A figure can exist having all the points in the line that bounds it equidistant from a single point in it.’

(b) ‘Any figure possessing this property is called a “circle”.’

Of these, (a) is an assumption about a matter of fact, and (b) is a genuine definition. Let us look at a demonstration that is said to depend on this definition, and see which of the two propositions it really relies on. It starts with

(i) ‘About the centre A, draw the circle BCD.’

This relies on (a) the assumption that such a figure can drawn. It makes no difference to the demonstration whether that figure is called a ‘circle’ or not. Nothing would be lost if we replaced (i) by

(i’) ‘Through the point B, draw a line returning into itself, of which every point is equidistant from the point A.’

This gets rid of the definition (b) and rests wholly on (a). The circle having been drawn, let us proceed to the consequence:

(ii) ‘Since BCD is a circle, the radius BA is equal to the radius CA.’

BA is equal to CA not because BCD is a circle, but because BCD is a figure with equal radii. Our warrant for assuming that such a figure can be drawn is (a). There may be dispute about whether postulates such as (a) rests on intuition or on proof; but either way they are the premises the theorems depend on; and while they are retained it would make no difference to the certainty of geometrical truths if every definition in Euclid, and every technical term therein defined, were set aside.

Perhaps it is superfluous to dwell at such length on something that is nearly self-evident; but when an obvious distinction has been muddled by powerful intellects, it’s better to say too much than too little for the purpose of making such mistakes impossible in future. [For our purposes Mill has already said enough. What he offers next is a long discussion of a demonstration about dragons; it is like the demonstration about circles except that Mill now allows opponents to try defending themselves at certain points, and convincingly slaps down the defences.]

§6. This notion that demonstrative truths follow from definitions rather than from the postulates implied in them—why has it survived so long? One reason is that the postulates, even in the sciences that are thought to be supreme in demonstrative certainty, are not always exactly true. It is not true that a circle exists or can be drawn which has all its radii exactly equal. Such accuracy is ideal only; it isn’t something we can •find in nature, still less is it something we can •make. This made it hard for people to conceive that the most certain conclusions could rest on premises which—far from being certainly true—are certainly not true to the full extent asserted. When I come to discuss demonstration I’ll deal with this apparent paradox, showing that ·in each case· as much of the postulate is true as is required to support as much as is true of the conclusion. But philosophers who weren’t satisfied by this thesis, or to whom it hadn’t occurred, have thought they had to find in definitions something more certain—or at least more precisely true—than the implied postulate of the real existence of an object corresponding to the definition. And what they came up with was this:

A definition is a statement and analysis not of •the mere meaning of a word, or of •the nature of a thing, but of •the nature of an idea.

What they thought about the proposition ‘A circle is a plane figure bounded by a line all the points of which are at an equal distance from a given point within it’ is an assertion not •that any real circle has that property (which wouldn’t be exactly true) but •that we conceive a circle as having it—i.e. that our abstract idea of a circle is an idea of a figure with its radii exactly equal.

In line with this, it is said that the subject-matter of mathematics, and of every other demonstrative science, is not •things as they really exist but •abstractions of the mind. A geometrical line is a line without breadth; but no such line exists in nature; it’s merely a notion suggested to the mind by its experience of nature. The definition (it is said) is a definition of this mental line, not of any actual line: and it is only of the mental line, not of any line existing in nature, that the theorems of geometry are precisely true.

Even if this doctrine about the nature of demonstrative truth were correct (and in II.5 I shall try to prove that it isn’t), the conclusions that seem to follow from a definition don’t follow from the definition as such but from an implied postulate. . . . The definition postulates the real existence of an idea such as it is describing; it assumes that the mind can frame [see Glossary], or rather has framed, the notion of length without breadth and without any other sensible property. I can’t see that the mind can form any such notion; it can’t conceive length without breadth; all it can do is, when contemplating objects, to attend to their length while ignoring their other sensible qualities, thus discovering what properties can be predicated of them purely in virtue of their length. If this is right, then what is postulated in the geometrical definition of a line is the real existence not of length without breadth but merely of length—i.e. of long objects. This is quite enough to support all the truths of geometry, because every property of a geometrical line is really a property of all physical objects in so far as possessing length. But even what I hold to be the false doctrine about this doesn’t affect the conclusion that our reasonings are grounded on the matters of fact postulated in definitions, not on the definitions themselves. . . .

§7. Although definitions are properly of names and not of things, it doesn’t follow that definitions are arbitrary. How to define a name may be an inquiry not only of considerable difficulty and intricacy but involving considerations going deep into the nature of the things denoted by the name. For example the inquiries that form the subjects of Plato’s most important Dialogues: ‘What is rhetoric?’ (the Gorgias) and ‘What is justice?’ (the Republic). Also the question scornfully asked by Pilate, ‘What is truth?’, and the fundamental question with moral theorists in all ages, ‘What is virtue?’

It would be a mistake to represent these difficult and noble inquiries as aiming only to ascertain the conventional meaning of a name. They are inquiries to determine not so much what is as what should be the meaning of a name; and this, like other practical questions of terminology, can’t be solved unless we enter, sometimes very deeply, into the properties not merely of names but of the things named.

Although the meaning of every concrete general name resides in the attributes it connotes, the objects were named before the attributes. You can see this in the fact that in all languages abstract names are mostly compounds or other derivatives of the corresponding concrete names. Thus the first names to be used (after proper names) were connotative names; and in the simpler cases a distinct connotation was presumably present to the minds of those who first used the name, and was distinctly intended by them to be conveyed by it. The first person who used the word ‘white’ as applied to snow or to any other object presumably knew very well what quality he intended to predicate, and had in his mind a perfectly distinct conception of the attribute signified by the name.

But where the resemblances and differences on which our classifications are based are not so obvious and easy to pin down—and especially where they consist in a number of qualities, the effects of which are not easy to sort out and assign to their respective causes—names are often applied to nameable objects by people who have no distinct connotation in mind. They’re merely influenced by a general resemblance between the new object and some or all of the old familiar objects that they have been accustomed to call by that name. This, as I have shown [here], is the law which even the mind of the philosopher must follow when giving names to the simple elementary feelings of our nature; but where the things to be named are complex wholes, a philosopher isn’t content with noticing a general resemblance—he examines what the resemblance consists in, and gives the same name to things only if they are alike in definite respects. So ·even in these complex cases· the philosopher habitually uses general names with a definite connotation. But language wasn’t made by philosophers, and can’t be much repaired by them. In the minds of the real arbiters of language—·the common people·—general names connote little more than a vague over-all resemblance to the things that they were earliest (or most) accustomed to call by those names. This is especially true when the classes the names denote can’t be brought before the tribunal of the outward senses to be identified and sorted out.

When ordinary folk predicate ‘just’ or ‘unjust’ of any action, ‘noble’ or ‘mean’ of any feeling, expression or demeanour, ‘statesman’ or ‘charlatan’ of anyone figuring in politics, do they mean to affirm of those various subjects any definite attributes of some kind? No! They merely recognise (or think they do) some fairly vague and loose likeness between these and some other things that they have been accustomed to label or hear labelled by those words.

Language, as Sir James Mackintosh used to say of governments, ‘is not made, but grows’. A name isn’t imposed on a class of objects because someone decided to do this; rather, it is applied first to one thing and then to another and another. . . . By this process it quite often happens that a name passes through successive links of resemblance from one object to another. . . until it comes to be applied to things that have nothing in common with the first things it was given to—though they retain the name also. Eventually the name denotes a confused jumble of objects that have nothing in common, so that it connotes nothing, not even a vague and general resemblance. When a name has fallen into this state, in which by predicating it of any object we assert literally nothing about the object, it is no longer fit for thought or the communication of thought, and can be made serviceable only by stripping it of some part of its multifarious denotation and confining it to objects with some attributes in common. These are the inconveniences of a language that ‘is not made, but grows’. Like governments, it can be compared to a road that has made itself: it requires continual mending in order to be passable.

This shows why there is often so much trouble about the definition of an abstract name. The question ‘What is justice?’ is equivalent to ‘What is the attribute that mankind mean to predicate when they call an action “just”?’ The answer to that is that they, having come to no precise agreement on the point, don’t mean to predicate distinctly any attribute at all. Yet they all believe that there’s some common attribute possessed by all the actions that they ordinarily call ‘just’. So the question has to be: ‘Is there any such common attribute?’ ·Really, there are three questions·:

(1) Do mankind agree sufficiently about the particular actions they call ‘just’ make it possible to ask what quality those actions have in common? If it is, then

(2) do those actions really have any quality in common? If they do, then

(3) what quality is it?

Of these three, only (1) is an inquiry into usage and convention; the other two are inquiries into matters of fact. And if (2) is answered negatively, there remains this:

(4) What is the best way to form, artificially, a class that the name may denote?

This is often more arduous than all the rest.

The study of the spontaneous growth of languages is of great importance to those who want to remodel them so as to make them more logical. The classifications roughly made by established language, when they are retouched (as they almost all need to be) by the hands of the logician, are often excellently suited to his purposes. As compared with the philosopher’s classifications they are like the customary law of a country, which has grown up spontaneously, compared with laws organised and digested into a code. The customary law is a less perfect instrument than the codified one; but being the result of a long. . . .course of experience, it contains a mass of materials that may be very helpful in the construction of a systematic written code. . . . When a name’s meaning is widened and widened until it is applied to things among which there isn’t even a rough superficial resemblance, at every step in its progress we shall find such a resemblance. And these changes in the meaning of words are often a pointer to real connections between the things denoted by them—connections that might otherwise escape the notice of thinkers. . . . The history of philosophy abounds in examples of such oversights when philosophers didn’t see the hidden link connecting the seemingly disparate meanings of some ambiguous word.[1]

Whenever the inquiry into the definition of the name of any real object consists of anything more than a mere comparison of authorities, we silently assume that we must find for the name a meaning that is •compatible with its continuing to denote all—or anyway the greater or the more important part—of the things it is commonly applied to. So the inquiry into the definition is an inquiry into the resemblances and differences among those things:

•Is there any resemblance running through them all? If not,

•through what portion of them can a general resemblance be traced? And lastly

•what are the common attributes that has given to them all, or to that portion of them, the resemblance that has led to their being classed together?

When these common attributes have been located and classified, the name that belongs in common to the resembling objects acquires a distinct connotation in place of the previous vague one; and by having this distinct connotation it becomes definable.

When the philosopher is giving the general name a distinct connotation, he’ll try select the most important of the attributes that are common to all the things usually denoted by the name; important either •in themselves or •indirectly through facts about their consequences—how many of them there are, or how conspicuous or interesting they are. He’ll do his best to select the differentiae that lead to the largest number of interesting propria [= plural of ‘proprium’]. Why? Because •these do a good job—better than •the more obscure and difficult qualities that they often depend on—of presenting a set of objects in a way that makes the groups into which they naturally fall look natural. But to penetrate to the more hidden agreement on which these obvious and superficial agreements depend is often one of the •most difficult of scientific problems, and usually one of the •most important. And because the result of this inquiry into the causes of the properties of a class of things has as a by-product a decision about what some word should mean, some of the deepest and best investigations that philosophy presents to us started as inquiries into the definition of a name, and have disguised themselves as that all through.

NOTES

[1] In an earlier writing of mine I said: ‘Few people have thought about how much knowledge of things is required to enable a man to say that an argument turns wholly upon words. Just about every leading philosophical term is used with almost innumerable shades of meaning, to express ideas more or less widely different from one another. Between two of these ideas a wise and penetrating mind will see, as it were intuitively, an unobvious connection; he may be unable to give a logical account of it, but he’ll base a perfectly valid argument on it—an argument which a less insightful critic will take to be a fallacy turning on the double meaning of a term. And the greater the genius of the one who safely leaps over the chasm, the greater will probably be the crowing and vainglory of the mere logician who, hobbling after him, displays his own superior wisdom by pausing on its brink, and giving up as hopeless his proper business of bridging it.’

Book II: Reasoning

Glossary

Chapter 1. Inference of reasoning in general

Chapter 2. Reasoning, or syllogism

Chapter 3. The functions and logical value of the syllogism

Chapter 4. Trains of reasoning, and deductive sciences

Chapter 5. Demonstration, and necessary truths

Chapter 6. Demonstration, and necessary truths (cont'd)

Chapter 7. Examining some of the opposition to the preceding doctrines

Glossary

A&NP: Acronym of the ‘all and nothing principle’, to which Mill refers only by its Latin title, dictum de omni et nullo. Explained here.

art: In this work, ‘art’ is a vehicle for several related ideas: rules, skill, techniques.

assertion: Mill uses this in about the way we use ‘proposition’. For there to be an ‘assertion’, in his sense, nobody needs to have asserted anything.

basic: This replaces Mill’s ‘original’ in some of its occurrences.

begging the question: Sometimes (not always) this replaces the Latin petitio principii. Mill’s sense of this phrase is the only sense it had until fairly recently: ‘beg the question’ was to offer a ‘proof’ of P from premises that include P. It now means ‘raise the question’ (‘That begs the question of what he was doing on the roof in the first place.’) It seems that complacently illiterate journalists (of whom there are many) encountered the phrase, liked it, guessed at its meaning, and plunged ahead without checking.

cognition: Cognitions are items of knowledge, in a weak sense of ‘knowledge’ such that a cognition doesn’t have to be true. In the context here they aren’t significantly different from beliefs.

connoting: To say that word W connotes attribute A is to say that the meaning of W is such that it can’t apply to anything that doesn’t have A. For example, ‘man’ connotes humanity.

identical proposition: Strictly speaking, this is a proposition of the form ‘x is x’, where the subject and predicate are identical. But the phrase came also to be used for any proposition where the meaning of the predicate is a part of the meaning of the subject.

import: In Mill’s use of it, this means about the same as ‘meaning’; but he does use both those words, and the present version will follow him in that.

meaning: In most places this is the word Mill has used, but sometimes it replaces his ‘acceptation’. It sometimes appears in the singular though the plural would seem more natural; that’s how Mill wrote it.

modify: To ‘modify’ a description is to amplify it adjectivally or adverbially, e.g. modifying ‘man’ with ‘irritable’, and ‘run’ with ‘swiftly’.

mutatis mutandis: A Latin phrase that is still in current use. It means ‘(mutatis) with changes made (mutandis) in the things that need to be changed’. The use of it implies that it’s obvious what the needed changes are.

noumenon: A Greek word, much used by Kant, meaning ‘thing considered as it is in its own nature’ in contrast with ‘thing considered in terms of how it appears’, i.e. phenomenon. The plural is noumena.

popular: Even as late as Mill’s time this mainly meant ‘of the people’, usually the not highly educated or very intelligent people. It didn’t mean ‘liked by the people’.

real: Here the word ‘real’ is tightly tied to its origin in the Latin res = ‘thing’. So the contrast between ‘real’ propositions and ‘verbal’ ones involves the contrast between things and words.

reductio ad absurdum: Standard (Latin) name for either of two forms of argument. (i) Proving P by showing that not-P logically implies P. (ii) Proving P by showing that not-P logically implies some Q that is obviously and indisputable false.

science: Any intellectual discipline whose doctrines are are highly organised into a logical structure. It doesn’t have to involve experiments, or to be empirical. Many philosophers thought that theology is a science.

signification: This seems to mean about the same as ‘meaning’, but Mill uses both words, and this version will respect his choices.

universal type of. . . : The basic central paradigm of. . . .

vortices: Plural of ‘vortex’. According to Descartes’s highly speculative astronomy, each planet was nested in a band of matter—a vortex—circling around the sun.

vulgar: Applied to people who have no social rank, are not much educated, and (the suggestion often is) not very intelligent.

Chapter 1. Inference of reasoning in general

§1. The topic of Book I was not the nature of proof but the nature of assertion [see Glossary]: the import conveyed by a proposition, whether or not it is true, not the means by which to distinguish true propositions from false ones. The proper subject of logic is proof, but before we could understand what proof is we had to understand •what it is that gets proved, •what it is that can be a subject of belief or disbelief, of affirmation or denial. in short, •what the different kinds of propositions assert.

I have pushed this preliminary inquiry far enough to get a definite result. (i) Assertion relates either to the meaning of words or to some property of the things that words signify. (ii) Assertions about the meaning of words, among which definitions are the most important, have an indispensable role in philosophy. (iii) But because the meaning of words is essentially arbitrary, this class of assertions can’t be true or false, and so can’t be proved or disproved. (iv) Assertions respecting things, or what may be called real [see Glossary] propositions, as against verbal ones, are of various sorts. I have analysed the import [see Glossary] of each sort, and have ascertained the nature of the things they relate to and the nature of what they say about those things. (v) I showed that whatever the form is of a proposition, and whatever its ostensible subject or predicate, the real subject of each proposition is •one or more facts or phenomena of consciousness, or •one or more of the hidden causes or powers to which we ascribe those facts; and that what is asserted or denied concerning those phenomena or powers is always either existence, order in place, order in time, causation, or resemblance. That’s the theory of the import of propositions reduced to its ultimate elements: but there’s a simpler way of putting it that doesn’t dig so deep but is scientific enough for many of the purposes for which such a general expression is required. . . . It goes like this:

Every proposition asserts that some given subject does (or that it doesn’t) have some attribute; or that some attribute is (or that it isn’t) conjoined with some other attribute in all or some of the subjects that have it.

Let us now move on to the special problem of the science [see Glossary] of logic, namely ‘How are assertions proved or disproved?’ This is being asked about propositions that are appropriate subjects of (dis)proof, not about ones that can be known through direct consciousness, i.e. intuition.

We say of a fact or statement that it is ‘proved’ when our belief in its truth is based on some other fact or statement from which it is said to follow. Most propositions. . . .that we believe are believed not •because they are obviously true but •because we think they can be inferred from something we have already accepted. Inferring a proposition from a previous proposition—giving credence to it or claiming credence for it as a conclusion from something else—is reasoning, in the most extensive sense of the term. There’s a narrower sense in which ‘reasoning’ is confined to the form of inference known as ‘ratiocination’—of which syllogism is the general type [see Glossary]. Early in Book I reasons were given for not using ‘reason(ing)’ in this restricted sense, and additional motives will be suggested by the considerations that I am now embarking on.

§2. Before coming to real inference, I should say a little about kinds of apparent inference, getting them out of the way so that they aren’t confused with the real thing. I’ll discuss four of them.

(a) The first sort occurs when the proposition Q that is ostensibly inferred from another proposition P turns out under analysis to be merely a repetition of all or a part of the assertion contained in P. All the textbook examples of equipollency—i.e. equivalence—of propositions are of this kind. Thus, if we were to argue

•Every man is rational; therefore no man is incapable of reason,

•No man is exempt from death; therefore all men are mortal,

it would be obvious that we weren’t proving anything but merely offering two wordings for a single proposition. One wording may have some advantages over the other, but it doesn’t offer a shadow of proof.

(b) Secondly, from a universal proposition we pretend to infer another that differs from it only in being particular:

•All A is B, therefore Some A is B;

•No A is B, therefore Some A is not B.

Here again we aren’t inferring one proposition from another, but merely asserting something and then repeating part of it.

(c) A third sort: From a proposition that affirms a predicate of a given subject we ‘infer’ a proposition affirming of the same subject something connoted by the former predicate— e.g. ‘Socrates is a man, therefore Socrates is a living creature’, where everything connoted by ‘living creature’ was affirmed of Socrates when he was said to be ‘a man’. (If the propositions are negative, we must reverse their order: ‘Socrates is not a living creature, therefore he is not a man’.) These are not really cases of inference; yet the trivial examples by which logic textbooks illustrate the rules of the syllogism are often of this ill-chosen kind—formal ‘demonstrations’ of conclusions to which anyone who understands the words in the premises has already consciously assented.

(d) The most complex case of this sort of apparent inference is ‘conversion’ of propositions: turning the predicate into a subject, and the subject into a predicate, and making out of the same terms thus reversed another proposition that must be true if the former is true. Thus, from the particular affirmative proposition Some A is B we may infer Some B is A. From the universal negative No A is B we may infer No B is A. From the universal affirmative proposition All A is B it can’t be inferred that all B is A, but it can be inferred that some B is A. . . . From Some A is not B we can’t even infer that some B is not A—some men are not Englishmen but it doesn’t follow that some Englishmen are not men. The only recognised way of converting such a particular negative proposition is by changing Some A is not B to Some A is a-thing-that-is-not-B; this is a particular affirmative, which can be simply converted to Some thing that is not B is A. . . .

In all these cases there’s no real inference; the conclusion presents no new truth, nothing but what was already •asserted in the premise and •obvious to anyone who understands it. The fact asserted in the conclusion is all or a part of the fact asserted in the premise. [Mill explains and defends this in terms of his Book I account of the import of propositions. One bit of this will be enough:] When we say that some lawful sovereigns are tyrants, what do we mean? That the attributes connoted by ‘lawful sovereign’ and the attributes connoted by ‘tyrant’ sometimes co-exist in one person. Now this is also precisely what we mean when we say that some tyrants are lawful sovereigns! So the latter isn’t a second proposition inferred from the first, any more than the English translation of Euclid’s Elements is a collection of theorems that are different from those contained in the Greek original—different from them and inferred from them. . . .

[In a footnote Mill explains some technical terms that won’t be needed in the rest of the work, except for the next paragraph. They are:

Contraries:

All A is B

No A is B

Subcontradictories:

Some A is B

Some A is not B

Contradictories:

All A is B

Some A is not B

”

No A is B

Some A is B

Subalternate:

All A is B

Some A is B

”

No A is B

Some A is not B]

Although you can’t call it ‘reasoning’ or ‘inference’ when something that is asserted is then asserted again in different words, it is extremely important to develop a skill in spotting, rapidly and accurately, cases where a single assertion is ·showing up twice·, disguised under diversity of language. And the cultivation of this skill falls strictly within the province of the art [see Glossary] of logic. That is the main function •of any logical treatise’s important chapter about the ‘opposition’ of propositions, and •of the excellent technical language logic provides for distinguishing the different kinds of opposition. Such considerations as these:

•Contrary propositions can both be false but can’t both be true;

•Subcontrary propositions can both be true but can’t both be false;

•Of two contradictory propositions one must be true and the other false;

•Of two subalternate propositions the truth of the universal proves the truth of the particular, and the falsity of the particular proves the falsity of the universal, but not vice versa;

are apt to appear at first sight to be very technical and mysterious; but when they’re explained they seem almost too obvious to need to be stated so formally. . . . In this respect, however, these axioms of logic are on a level with those of mathematics. Things that are equal to the same thing are equal to one another—this is as obvious in any particular case as it is in the general statement; and if no such general maxim had ever been laid down, the demonstrations in Euclid would never have been stopped in their tracks by the gap which is at present bridged by this axiom. Yet no-one has ever censured writers on geometry for putting a list of these elementary generalisations at the start of their treatises as a first exercise of the ability to grasp a general truth, this being something the learner needs at every step. And the student of logic, in the discussion even of such truths as are cited above, acquires habits of •wary interpretation of words and of •exactly measuring the length and breadth of his assertions. Such habits are among the most indispensable conditions of any considerable mental attainment, and it’s one of the primary objects of logical discipline to cultivate them.

§3. . . . .Let us now move on to cases where the progression from one truth to another really does involve inference in the proper sense of the word—-ones where we set out from known truths to arrive at others that are really distinct from them.

Reasoning, in the extended sense in which I use the word, in which it is synonymous with ‘inference’, is commonly said to be of two kinds:

(1) induction: reasoning from particular propositions to general ones,

(2) ratiocination or syllogism: reasoning from general propositions to particular ones.

I shall show that there’s a third species of reasoning that doesn’t fit either of those descriptions but is nevertheless valid and is indeed the foundation of both the others.

I have to point out that the expressions ‘reasoning from particular propositions to general ones’ and ‘reasoning from general propositions to particular ones’ don’t adequately mark the distinction between •induction (in the sense I am giving it) and •ratiocination—or anyway they don’t mark it without the aid of a commentary. [We are to understand, it seems, that the required ‘commentary’ is the rest of this paragraph.] What these expressions mean is that induction is inferring a proposition from propositions less general than itself, and ratiocination is inferring a proposition from propositions equally or more general. When, from the observation of a number of instances we ascend to a general proposition, or when by combining a number of general propositions we conclude from them another proposition still more general, the process—which is substantially the same in both cases—is called ‘induction’. When from a general proposition combined with other propositions we infer a proposition of the same degree of generality as itself, or a less general proposition, or a merely individual proposition, the process is ratiocination. ·Why ‘combined with other propositions’? Because· from a single proposition nothing can be concluded that isn’t involved in ·the meanings of· the terms. . . .

Given that all experience begins with individual cases and proceeds from them to general propositions, it might seem that the natural order of thought requires that induction should be treated of before we reach ratiocination. But in a science that aims to trace our acquired knowledge to its sources, it is best that the inquirer should start with the later rather than the earlier stages of the process of constructing our knowledge—tracing derivative truths back to the truths they •are deduced from and •depend on for their believability—before trying to pin-point the spring from which both ultimately take their rise. There’s no need for me to justify or explain this here; the advantages of this order of proceeding will show themselves as we advance.

So all I’ll say about induction here is that it is without doubt a process of real inference. The conclusion in an induction takes in more than is contained in the premises. The principle or law collected from particular instances—the general proposition in which we embody the result of our experience—covers a much bigger territory than the individual experiments on which it is based. A principle arrived at on the basis of experience is more than a mere summing up of individual observations; it’s a generalisation •grounded on those cases and •expressive of our belief that what we found true in them is true in indefinitely many cases that we haven’t examined and probably never will. The nature and grounds of this inference, and the conditions necessary to make it legitimate, will be the topic of Book III; but it can’t be doubted that such inference really does take place. . . .

So induction is a real process of reasoning or inference. Whether, and in what sense, as much can be said of the syllogism remains to be decided by the examination that begins now.

Chapter 2. Reasoning, or syllogism

§1. The analysis of the syllogism has been so accurately and fully performed in the common logic textbooks that the present work, which is not designed as a textbook, needs only to recapitulate the leading results of that analysis, as a basis for what I’ll say later about the functions of the syllogism and the place it holds in science.

In a legitimate syllogism there have to be exactly three propositions—the proposition to be proved (the conclusion) and two other propositions which together prove it (the premises). There must be exactly three terms—the subject and predicate of the conclusion, and the ‘middle term’, which must occur in both premises because its role is to connect the other two terms. The predicate of the conclusion is called the major term of the syllogism; the subject of the conclusion is called the minor term. Each of these must occur in just one of the premises, together with the middle term which occurs in both. The premise containing the major term is called the major premise; that which contains the minor term is called the minor premise.

Syllogisms are divided by most logicians into four ‘figures’. . . .according to the position of the middle term, which may either be the subject in both premises, the predicate in both, or the subject in one and the predicate in the other. The most common case is that in which the middle term is the subject of the major premise and the predicate of the minor. This is reckoned as the first figure. When the middle term is the predicate in both premises, the syllogism belongs to the second figure; when it is the subject in both, to the third. In the fourth figure the middle term is the subject of the minor premise and the predicate of the major. . . .

[The following schema provides a simple way of remembering what each of the ’figures’ is:

First

Second

Third

Fourth

M—C

C—M

M—C

C—M

A—M

A—M

M—A

M—A

Draw a line through M, sloping down, then up, then down, then sloping up: the result is a W. You’ll have little need for this as you read on, and even less for the stuff on this page and the next about the ‘moods’ of the syllogistic figures. Its inclusion here is mere act of piety towards Mill.]

Each figure is divided into moods, according to what are called the propositions’ quantity (i.e. whether they are universal or particular) and their quality (i.e. whether they are affirmative or negative). Here are schemas for all the moods in which the conclusion does follow from the premises. A is the minor term, C the major, M the middle term.

FIRST FIGURE

All M is C

No M is C

All M is C

No M is C

All A is M

All A is M

Some A is M

Some A is M

∴

∴

∴

∴

All A is C

No A is C

Some A is C

Some A is not C

SECOND FIGURE

No C is M

All C is M

No C is M

All C is M

All A is M

No A is M

Some A is M

Some A is not M

∴

∴

∴

∴

No A is C

No A is C

Some A is not C

Some A is not C

THIRD FIGURE

All C is M

No M is C

Some M is C

All M is C

Some M is not C

No M is C

All M is A

All M is A

All M is A

Some M is A

All M is A

Some M is A

∴

∴

∴

∴

∴

∴

Some A is C

Some A is not C

Some A is C

Some A is C

Some A is not C

Some A is not C

FOURTH FIGURE

All C is M

All C is M

Some C is M

No C is M

No C is M

All M is A

No M is A

Some M is A

All M is A

Some M is A

∴

∴

∴

∴

∴

Some A is C

Some A is not C

Some A is C

Some A is not C

Some A is not C

In these blank forms for making syllogisms, no place is assigned to singular propositions. They are of course used in ratiocination; but because their predicate is affirmed or denied of the whole of the subject they are ranked for the purposes of the syllogism with universal propositions. So these two syllogisms—

•All men are mortal, All kings are men, therefore All kings are mortal

•All men are mortal, Socrates is a man, therefore Socrates is mortal

—are precisely similar arguments and are both ranked in the first mood of the first figure. [Mill has here an enormous footnote critically discussing Bain’s view that singular propositions don’t belong in syllogisms. He argues convincingly that Bain’s case for this rests on assuming that proper names have meanings, although elsewhere in his work he affirms Mill’s view that they don’t.]

If you want to know why the above forms are legitimate and that no others are, you could probably work that out for yourself, or learn it from just about any ordinary common-school book on syllogistic logic, or you could go to Whately’s Elements of Logic, where the whole of the common doctrine of the syllogism is stated with philosophical precision and explained with remarkable clarity.

All valid ratiocination—all reasoning from admitted general propositions to other propositions equally or less general—can be exhibited in some of the above forms. The whole of Euclid, for example, could easily be expressed in a series of syllogisms, regular in mood and figure.

Though a syllogism fitting any of these formulae is a valid argument, any correct ratiocination can be stated in syllogisms of the first figure. The rules for putting an argument in one of the other figures into the first figure are called rules for the ‘reduction’ of syllogisms. It is done by converting one or both of the premises. Thus an argument in the first mood of the second figure—

No C is M

All A is M

∴

No A is C,

can be reduced as follows. The premise No C is M can be replaced by No M is C, which I have shown to be the very same assertion in other words. With that change made, the syllogism becomes

No M is C

All A is M

∴

No A is C,

which is a good syllogism in the second mood of the first figure. It’s equally easy to reduce a syllogism in the first mood of the third figure—

All M is C

All M is A

∴

Some A is C,

—to one in the third mood of the first figure—

All M is C

Some A is M

∴

Some A is C.

That involves replacing ‘All M is A’ by ‘Some A is M’; that’s not the same proposition, but it asserts a part of what ‘All M is A’ asserts, and that part suffices to prove the conclusion. Similar moves enable every mood of every other figure to be reduced to one or other of the moods of the first figure; those with affirmative conclusions reduce to the first or third moods of the first figure, those with negative conclusions reduce to the second or fourth. . . .

Sometimes an argument falls more naturally into one of the other three figures, with its conclusiveness being more immediately obvious in some figure other than the first. Compare this third-figure syllogism

Aristides was virtuous,

Aristides was a pagan, therefore

Some pagan was virtuous

with the first-figure syllogism that it reduces to:

Aristides was virtuous,

Some pagan was Aristides, therefore

Some pagan was virtuous.

The third-figure version of the argument is more natural, and more immediately convincing, than the first-figure version.

[Mill mentions a 1764 account by the German philosopher Johann Heinrich Lambert of the purposes for which each syllogistic figure is most natural. Although he is respectful towards this work of Lambert’s, Mill concludes:] We are at liberty, in conformity with the general opinion of logicians, to consider the two elementary forms of the first figure as the universal types [see Glossary] of all correct ratiocination—one when the conclusion to be proved is affirmative, the other when it is negative. [One of the ‘two elementary forms’ is the first and third moods, the other is the second and fourth moods.] Some arguments may have a tendency to clothe themselves in the forms of the second, third, and fourth figures; but this can’t possibly happen with the only arguments that are of first-rate scientific importance, namely those in which the conclusion is a universal affirmative, because such conclusions can be proved only in the first figure.[1]

§2. On examining, then, these two general formulae, we find that in both of them, one premise, the major, is a universal proposition; and according as this is affirmative or negative, the conclusion is so too. All ratiocination, therefore, starts from a general proposition, principle or assumption in which a predicate is affirmed or denied of an entire class. . . .i.e. of an indefinite number of objects distinguished by a common characteristic and on that basis designated by a common name. [Remember that what makes this the ‘major’ premise isn’t its •being written first but its •containing the predicate of the conclusion.]

The other (‘minor’) premise is always affirmative, and asserts that something—an individual or a part or all of a class—belongs to the class of which something was affirmed or denied in the major premise. So the attribute affirmed or denied of the entire class may (if that affirmation or denial was correct) be affirmed or denied of the object(s) said to be included in the class; which is just what the conclusion asserts.

Is that an adequate account of the constituent parts of the syllogism? We’ll soon see. But it is at least true as far as it goes. It has accordingly been generalised, and erected into the logical maxim that whatever can be affirmed or denied of a class may be affirmed or denied of everything included in the class. This so-called ‘all and nothing principle’ is said to be the basis for all ratiocination—so much so that the answer to ‘What is ratiocination?’ is said to be ‘Applying the all and nothing principle’. [Mill gives the principle its standard Latin name, Dictum de omni et nullo. The present version will use the English name, usually abbreviated to ‘A&NP’.]

But this maxim, considered as a principle of reasoning, seems suited to a metaphysic that •was once generally accepted but •has for the last two centuries been considered as finally abandoned (though even today there are attempts to revive it). I’m talking about the metaphysical view that

what are called ‘universals’ are substances of a special kind, having an objective existence distinct from the individuals that are classified in terms of them.

If that were right, the A&NP would convey an important meaning. ·According to the dead metaphysical view about the nature of universals, we should think of ‘All men are rational’ as meaning ‘Man is rational’, where ‘Man’ stands for a substantial universal that has a certain relation R to each individual man. Then it would be a solid bit of news that the rationality involved in the nature of Man is also involved in the nature of each thing to which Man has the relation R, i.e. of each man·. That everything predicable of the universal is predicable of the various individuals contained under it is not an identical proposition [see Glossary] but a statement of a fundamental law of the universe. The assertion that the entire nature of the substantial universal forms part of the nature of each individual substance called by the same name—that the properties of Man, for example, are properties of all men—was a proposition of real significance when ‘Man’ did not mean all men but something inherent in men and vastly superior to them in dignity. But now that we know that

•a class—a universal, a genus or species—is not an entity in its own right but merely the individual substances that are placed in it, and that

•there’s nothing real in this situation except those substances, a common name given to them, and common attributes indicated by the name,

please tell me what we learn by being told that whatever can be affirmed of a class can be affirmed of every object in it! The class is nothing but the objects contained in it, and the A&NP amounts to the identical proposition that whatever is true of certain objects is true of each of them. [The crucial point here is Mill’s rejection of what he sees as the dead metaphysical view that when a substance has a certain property this involves two things and a relation between them rather than one thing that is thus-and-so.] If all ratiocination were merely the application of this maxim to particular cases, the syllogism would indeed be ‘solemn trifling’, which it has often been accused of being. The A&NP is on a par with another truth that also used to be regarded as highly important, namely ‘Whatever is, is’. To give any real meaning to the A&NP we must regard it not as an axiom but as a definition; we must look on it as intended to be a round-about account of the meaning of the word ‘class’.

An error that seemed to be finally refuted and dislodged from thought often needs only put on a new suit of phrases to be welcomed back to its old lodgings, and allowed to rest unquestioned for another cycle of years. Modern philosophers have been ruthless in expressing their contempt for the scholastic dogma that:

Genera and species are a special peculiar kind of substances—general substances that are the only permanent things—while the individual substances that come under them are continually changing; so that knowledge, which necessarily brings stability, must concern those general substances or universals, and not the facts or particulars that come under them.

Yet this nominally rejected doctrine has never ceased to poison philosophy. It has done this under the guise •of ‘abstract ideas’ in the work of Locke (though this has been less spoiled by it than the work of any other writer who has been infected with it), •of the ultra-nominalism of Hobbes and Condillac, or •of the ontology of the later German metaphysicians.

Once men got used to thinking of scientific investigation as essentially a study of universals, they didn’t drop this habit of thought when they stopped thinking of universals as having an independent existence. Even those who came to regard universals as mere names couldn’t free themselves from the notion that the investigation of truth is at least some kind of conjuration or juggle with those names. [In that striking phrase Mill is suggesting something like pulling a rabbit out of a hat.] When a philosopher •accepted the nominalist view of the signification [see Glossary] of general language also •accepted the A&NP [see Glossary] as the basis of all reasoning, those two premises committed him to some rather startling conclusions! Some writers who were deservedly celebrated held that the process of •arriving at new truths by reasoning consists merely in •substituting of one set of arbitrary signs for another—a doctrine that they think is conclusively confirmed by the example of algebra. . . . The culminating point of this absurd philosophy is Condillac’s aphorism that a science is almost nothing but une langue bien faite—i.e. that way to discover the properties of objects is to name them properly! The truth of course is the reverse of that: you can’t name things properly until you know what their properties are. . . . Common sense holds—and philosophical analysis confirms this—that the function of names is only to enable us to remember and communicate our thoughts. It’s true that they also enormously increase the power of thought itself, but there’s nothing mysterious about how they do this. They do it by the power inherent in an artificial memory, an instrument whose immense potency has been largely neglected. As an artificial memory, language truly is what it is often called, namely an instrument of thought; but it’s one thing to be the instrument and another to be the exclusive subject on which the instrument is exercised!. . . . There can’t be a greater error than to imagine that thought can be carried on with nothing in our mind but names, or that we can make the names think for us.

§3. Those who considered the A&NP as the foundation of the syllogism had a view of •arguments that corresponded to Hobbes’s wrong view about propositions (see I.5.2). Because some propositions are merely verbal, Hobbes—apparently wanting a definition that would cover all the cases—defined ‘proposition’ in a way implying that no proposition declares anything except the meaning of words. If he were right about this—if that’s all that could be said about the import of propositions—the theory we’d have to accept about what happens in a syllogism is the commonly accepted one. If the minor premise says only that something A belongs to class M, and the major premise says only that M is included in another class C, the conclusion would be only that whatever is in A is also in C; which tells us only that the classification is consistent with itself. But we have seen that there’s more to the meaning of a proposition than its merely putting something into or out of a class. Every proposition that conveys real [see Glossary] information asserts a matter of fact that depends not on classification but on the laws of nature. It asserts that a given object does/doesn’t have a given attribute; or it asserts that two attributes or sets of attributes do/don’t always or sometimes co-exist. . . . Any theory of ratiocination that doesn’t respect this import of propositions can’t possibly be the true one.

Applying this view of propositions to the two premises of a syllogism, here’s what we get. The major premise (which, remember, is always universal) says that all things that have a certain attribute (or attributes) A_1 do/don’t also have a certain other attribute (or attributes) A_2. The minor premise says that the thing or set of things which are the subject of that premise have A_1; and the conclusion is that they do/don’t also have A_2. Thus in our former example,

All men are mortal,

Socrates is a man, therefore

Socrates is mortal,

the subject and predicate of the major premise are connotative terms, denoting objects and connoting [see Glossary] attributes. The assertion in the major premise is that the attributes connoted by ‘man’ are always conjoined with the attribute called ‘mortality’. The minor premise says that the individual named ‘Socrates’ has the former attributes; and the conclusion is that he also has the attribute mortality. [Mill then goes through it again, with ‘Socrates is’ replaced by ‘All kings are’.]

If the major premise is negative, e.g. ‘No men are omnipotent’, it says that the attributes connoted by ‘man’ never exist with the ones connoted by ‘omnipotent’; from which, together with the minor premise, it is concluded that the same incompatibility exists between the attribute omnipotence and those constituting a king. We can analyse any other syllogism in the same general way.

[In a footnote Mill explains that in this next paragraph ‘A_1 coexists with A_2’ means only that some one thing has both—not that it has them at the same time.] If we look for the principle or law involved in every such inference, and presupposed in every syllogism whose premises and conclusion aren’t merely verbal, what we find is not the unmeaning A&NP but two fundamental principles that strikingly resemble the axioms of mathematics.

(i) The principle of affirmative syllogisms: things that co-exist with the same thing co-exist with one another. Or, more precisely: a thing that co-exists with another thing, which in turn co-exists with a third thing, also co-exists with that third thing.

(ii) The principle of negative syllogisms: a thing that co-exists with another thing which does not co-exist with a certain third thing doesn’t itself co-exist with that third thing. These axioms plainly relate to •facts, not to •conventions; and one or other of them is the basis for the legitimacy of every argument in which facts and not conventions are the subject-matter.

[At this point Mill launches a very long footnote responding to Herbert Spencer’s criticism of this account of syllogisms. The criticism rests on the assumption that the attribute humanity that you have is like the attribute humanity that I have, but that it’s not the very same attribute. We needn’t go through Mill’s entire treatment of this, but one part of it ought to be given here. Namely:] The meaning of any general name is some outward or inward phenomenon, ultimately consisting of feelings. If the continuity of these feelings is for an instant broken, they are no longer the same feelings, in the sense of individual identity. What, then, is the common ‘something’ that gives a meaning to the general name? Spencer can only say that it is the similarity of the feelings. I reply that the attribute is precisely that similarity. The names of attributes are in the last analysis names for the resemblances of our sensations (or other feelings).

§4. I showed in I.6.5 that there are two languages in which we can express all propositions, and therefore all combinations of propositions; I have used one of the two in giving my account of the syllogism, and I should now show how to translate the account into the other language. One of the two is theoretical, the other practical:

•Theoretical: the proposition is regarded as a portion of our knowledge of nature: an affirmative general proposition asserts the speculative truth that whatever has a certain attribute also has a certain other attribute.

•Practical: the proposition is regarded as a memorandum for our guidance—not a part of our knowledge but an aid in our practical activities, enabling us when we learn that an object has attribute A_1 to infer that it also has A_2, thus employing A_1 as a mark or evidence of A_2.

[Mill might have sharpened the contrast between what is theoretical (or speculative) and what is practical by expressing the latter in terms of imperatives: ‘When you find that something has A_1, expect it to turn out also to have A_2.’]

With propositions looked at in the second way, every syllogism comes within the following general formula:

Attribute A_1 is a mark of attribute A_2,

The given object has the mark A_1, therefore

The given object has the attribute A_2.

For example:

The attributes of man are a mark of the attribute mortality,

Socrates has the attributes of man, therefore

Socrates has the attribute mortality.

[And Mill does something similar with the other two syllogisms he has presented—with ‘All kings’ replacing ‘Socrates’, and ‘are not omnipotent’ replacing ‘are mortal’.]

To correspond with this alteration in the form of the syllogisms, the underlying axioms ·stated a page back· must also be altered. In this altered phraseology, both those axioms can be brought under one general expression:

Whatever has any mark, has that of which it is a mark.

Or when both premises are universal:

Whatever is a mark_1 of any mark_2 is a mark of whatever mark_2 is a mark of.

To check that these mean the same as the previously state ones can be left to the intelligent reader. As we proceed we’ll find that this ·practical· phraseology is very convenient. It’s the best way I know of to express with precision and force what is aimed at, and what is actually accomplished, in every case where truth is learned by ratiocination.[2]

NOTES

[1] [This footnote originally discussed work by William Hamilton and by Augustus De Morgan. The former of these—about ‘the quantification of the predicate’—is omitted here, as a dead end. Some of the latter is retained because, despite Mill’s coolness about it, it did lead somewhere. Incidentally, these two writers later had a controversy about the ‘quantification of the predicate’, in which (according to C. S. Peirce) ‘the reckless Hamilton flew like a dor-bug into the brilliant light of De Morgan’s mind’.]

Since this chapter was written a treatise has appeared which aims at a further improvement in the theory of the forms of ratiocination, namely De Morgan’s Formal Logic; or the Calculus of Inference, Necessary and Probable. In the more popular [see Glossary] parts of this volume there’s an abundance of valuable observations felicitously expressed; but its the principal feature of originality is an attempt to bring within strict technical rules the cases where a conclusion can be drawn from premises of a form usually classified as ‘particular’. De Morgan rightly says that from the premises Most Ms are Cs and Most Ms are As it strictly follows that Some As are Cs, because two portions of the class M, each containing more than half, must have some overlap. Following out this line of thought, it is equally evident that if we knew exactly what proportion the ‘most’ in each of the premises bear to the entire class M, we could correspondingly increase the definiteness of the conclusion. If 60% of M are included in C, and 70% in A. . . . the number of As that are Cs must be ≥ 30% of the class M. Proceeding on this conception of ‘numerically definite propositions’, and extending it to such forms as these [details omitted by this version, not by Mill] and examining what inferences can be drawn from the various possible combinations of premises of this description, De Morgan establishes universal formulae for such inferences; creating for that purpose not only a new technical language but a formidable array of symbols analogous to those of algebra.

The inferences presented by De Morgan are legitimate, and the ordinary theory ·of syllogisms· doesn’t deal with them; so I don’t say that it wasn’t worthwhile to show in detail how they could expressed in formulae as rigorous as those of Aristotle. . . . But I doubt that these results of his are worth studying and mastering for any practical purpose. The practical use of technical forms of reasoning is to keep out fallacies; but in ratiocination properly so called the fallacies that threaten arise from the incautious use of ordinary forms of language, and the logician must track the fallacies into that territory, rather than waiting for them on his territory. While the logician remains among propositions with the numerical precision of the calculus of probabilities, his enemy is left in possession of the only ground on which he can be formidable. Very few of the non-universal propositions that a thinker has to depend on for purposes either of speculation or of practice, can be made numerically precise, so common reasoning can’t be translated into De Morgan’s forms, which therefore can’t throw any light on it.

[2] [This footnote began with a long response to a fairly weak criticism by Bain. A second ‘more fundamental objection’ of Bain’s is also discussed: it turns on whether Mill’s ‘practical’ axiom is fitting for what Bain calls ‘Deductive Reasoning’, which he says consists in ‘the application of a general principle to a special case’. ‘Anything that fails to make prominent this circumstance’, Bain says, ‘is not adapted as a foundation for the syllogism’, so the right fundamental axiom is A&NP. Mill says that Bain is stipulating an unduly narrow meaning for the phrase ‘deductive reasoning’; and he also counter-attacks: ‘If the A&NP makes prominent the fact of the application of a general principle to a particular case, the axiom I propose makes prominent the condition which alone makes that application a real inference.’ He continues:]

I conclude, therefore, that both forms have their value and their place in logic. The A&NP should be retained as the fundamental axiom of the logic of mere consistency, often called ‘formal logic’; and I have never quarreled with the use of it in that role, or proposed to banish it from treatises on formal logic. But the other is the proper axiom for the logic of the pursuit of truth by way of deduction; and you have to recognise it if you want to show how deductive reasoning can be a road to truth.

Chapter 3. The functions and logical value of the syllogism

§1. I have shown •what the real nature is of the truths that syllogisms deal with (against the common theory’s more superficial account of their import), and •what the fundamental axioms are on which the force or syllogisms depends. Our next question about •the syllogistic process, that of reasoning from generals to particulars, is this:

Is •it a process of inference? a progress from the known to the unknown? a means of reaching items of knowledge that we didn’t know before?

Logicians have been remarkably unanimous in their way of answering this question, ·or at least of implying an answer·. Everyone says that a syllogism is bad if there’s anything more in the conclusion than was assumed in the premises; but that’s equivalent to saying that a syllogism can’t prove anything that wasn’t already known or assumed. Are we to conclude, then, that ratiocination isn’t a process of inference? And that the syllogism—which has often been said to be the only genuine reasoning—isn’t really entitled to be called ‘reasoning’ at all? This seems to follow, and indeed everyone who writes about syllogisms accepts that a syllogism can’t prove anything not involved in its premises. Yet some writers who explicitly acknowledge this still hold •that the syllogism is the correct analysis of what the mind does when discovering and proving the bulk of the things we believe, in science and in daily life; while those who have avoided this inconsistency have been led to claim that •the syllogistic theory itself is useless and frivolous because of the begging of the question [see Glossary] that they allege to be inherent in every syllogism. I believe that both these opinions are basically wrong, and that the defenders and the attackers of the syllogistic theory seem to have overlooked (or barely glanced at) certain considerations that have to be taken seriously if we are to understand the true character of the syllogism and the functions it performs in philosophy.

§2. It must be granted that in every syllogism, considered as an argument to prove the conclusion, there is a begging of the question. When we say,

All men are mortal,

Socrates is a man, therefore

Socrates is mortal,

the enemies of the syllogistic theory are certainly right in saying •that the proposition Socrates is mortal is presupposed in All men are mortal; •that we can’t be sure of the mortality of all men unless we are already sure of the mortality of each individual man; •that any doubt we have about the mortality of Socrates (or anyone else we choose to name) creates the same amount of doubt regarding All men are mortal; •that the general principle, instead of being given as evidence of the particular case, can’t itself be accepted as true without exception until every shadow of doubt that could affect any individual case within it has been dispelled by evidence from some other source, leaving nothing for the syllogism to prove; •that (in short) no reasoning from general propositions to particular ones can prove anything, because the only particulars we can infer from a general principle are ones that the general principle itself assumes as known.

This doctrine appears to me indestructible. Logicians who couldn’t fault it have usually tried to explain it away—not because they found any flaw in the argument itself, but because the contrary opinion seemed to rest on equally indisputable arguments. In the above syllogism or in any of my other examples, isn’t it obvious that the conclusion may be a truth that is genuinely new to the person to whom the syllogism is presented? Isn’t it a daily experience that truths previously unthought of—facts that haven’t been and can’t be directly observed—are arrived at through general reasoning? We believe that the Duke of Wellington is mortal. We don’t know this by direct observation as long as he isn’t yet dead. If we are asked ‘Then how do you know he is mortal?’ we would probably answer ‘Because all men are mortal’. Here, therefore, we arrive at the knowledge of a truth that can’t yet be learned by observation, reaching it by reasoning that can be exhibited in the following syllogism:

All men are mortal,

The Duke of Wellington is a man, therefore

The Duke of Wellington is mortal.

And since much of our knowledge is acquired in this way, logicians have persisted in representing the syllogism as a process of inference or proof; though none of them has solved the problem of reconciling that assertion with the thesis that if there’s anything in the conclusion that wasn’t already asserted in the premises the argument is bad. We can’t attach any serious scientific value to the distinction drawn between being •involved by implication in the premises and being •directly asserted in them. When Whately says that the object of reasoning is ‘merely to expand and unfold the assertions wrapped up and implied in those with which we set out, and to bring a person to see the full force of what he has admitted’, he doesn’t meet the real problem that confronts him, namely explaining how a science like geometry can be all ‘wrapped up’ in a few definitions and axioms. Also, this defence of the syllogism doesn’t differ much from the accusation that its assailants urge against it when they charge it with being useless for pressing the consequences of an admission that an opponent has been trapped into accepting without having understood its full force. When you accepted the major premise you asserted the conclusion; but, says Whately, you asserted it merely by implication, which must mean that you asserted it unconsciously—that you didn’t know you were asserting it. But then the difficulty re-appears in this shape—Oughtn’t you to have known? Were you warranted in asserting the general proposition without having satisfied yourself of the truth of everything that it includes? And if not, isn’t the syllogistic art obviously what its attackers affirm it to be, a contrivance for catching you in a trap and holding you fast in it?[1]

§3. There seems to be just one way out of this difficulty. The proposition that the Duke of Wellington is mortal is evidently an inference, something reached as a conclusion from something else. But do we really conclude it from the proposition All men are mortal? I answer No.

I think the error has consisted in overlooking the distinction between two parts of the process of philosophising, the •inferring part and the •registering part; and ascribing to the latter the functions of the former. . . . If a person is asked a question that he can’t answer right now, he may refresh his memory by turning to a memorandum that he carries about with him. But if he were asked ‘How did you come to know that?’ he won’t answer ‘From its being written in my note-book’—unless the book was written, like the Koran, with a quill from the wing of the angel Gabriel!

Assuming that the Duke of Wellington is mortal is immediately inferred from All men are mortal, where do we get our knowledge of that general truth from? Of course from observation. Now, all we can observe are individual cases. All general truths must be drawn from •individual cases, and they can be analysed back into •these; for a general truth is merely an aggregate of particular truths, a comprehensive expression by which indefinitely many individual facts are affirmed or denied at once. But a general proposition is not merely a compendious form for recording and preserving in the memory a number of particular facts, all of which have been observed. Generalisation is not a process of mere naming, it is also a process of inference. From instances that we have observed we feel [Mill’s word] entitled to conclude that what we found true in those instances holds in all similar cases, past, present and future. We then use the valuable contrivance of language that enables us to speak of many as if they were one, and record all that we have observed, together with all that we infer from our observations, in one concise expression; and thus we have only one proposition to remember or to communicate, instead of an endless number of them. The results of many observations and inferences, and instructions for making innumerable inferences in unforeseen cases, are compressed into one short sentence.

Thus, when we conclude from the death of John and Thomas, and everyone else we ever heard of who has died, that the Duke of Wellington is mortal like the rest; we may indeed pass through All men are mortal as an intermediate stage; but the inference doesn’t occur in the latter half of the process, the descent from all men to the Duke of Wellington. The inference is finished when we have asserted that all men are mortal. All we have to do then is to decipher our own notes.

[This paragraph starts with a rejection of a rather obscure view of Whateley’s. We can safely rejoin Mill as he emerges from that episode:] If from our experience of John and Thomas and all the others who lived and then died we’re entitled to conclude that all human beings are mortal, surely we are entitled to conclude immediately from those instances that the Duke of Wellington is mortal. The mortality of John and the others is all the evidence we have for the mortality of the Duke of Wellington. Nothing is added to the proof by interpolating a general proposition. Since the individual cases are all the evidence we can have—evidence that can’t be strengthened by any choice of logical form for it—. . . .I can’t see why we should obey the arbitrary fiat of logicians who •forbid us to take the shortest route from these premises to the conclusion, and •require us to get there by travelling the ‘high priori road! I can’t see why it should be impossible to journey from one place to another unless we ‘march up a hill, and then march down again’. It may be the safest road, and at the top of the hill there may be a resting-place with a good view of the surrounding country; but for the mere purpose of arriving at our journey’s end, our taking that road is optional—it’s a question of time, trouble, and danger.

Reasoning from one particular proposition to another without passing through a general proposition—that is not only possible, it’s something we do all the time. All our earliest inferences are like that. We draw inferences as soon as we can think at all, and we don’t learn to use general propositions until years later. The child, who, having burned his fingers, avoids putting them into the fire a second time has reasoned or inferred, but he hasn’t thought of the general maxim Fire burns. He knows from memory that he has been burned, and on this evidence he believes that if he puts his finger into the flame of a candle he will be burned again. He believes this each time he encounters flame, but he isn’t looking beyond this present flame. He isn’t generalising; he’s inferring a particular from particulars. That is the way the lower animals reason. There’s no evidence that any of them can use signs of the sort that are needed to make general propositions possible. . . . Not only the burned child, but the burned dog, dreads the fire.

When we draw inferences from our personal experience and not from maxims handed down to us by books or tradition, I think we do this by going from particulars to particulars •directly, much oftener than by going •through the intermediate agency of a general proposition. We are constantly reasoning from ourselves to other people, or from one person to another, without taking the trouble to erect our observations into general maxims about human or external nature. When we conclude that ‘That’s how he will act when. . . etc.’ we are less likely to be •relying on some general view about people like him, or about people generally, than to be merely •recollecting his feelings and conduct on some previous occasion or •considering how we would feel or act ourselves. The village matron who is consulted about the health of a neighbour’s child identifies the illness and its remedy simply by remembering what she regards as the similar case of her Lucy. And we all guide ourselves in the same way when we have no definite maxims to steer by; and if we have had an extensive experience, and strongly retain our impressions of it, we can in this manner acquire a considerable power of accurate judgment, without being able to justify it or communicate it to others. . . . An old warrior, after a rapid glance at the outlines of the ground, can immediately give the necessary orders for a skillful arrangement of his troops; and if he hasn’t had much theoretical instruction or often been required to answer to other people for his conduct, he may never have had in his mind a single general theorem about the relation between terrain and deployment. His experience of encampments in somewhat similar circumstances has left a number of vivid, unexpressed, ungeneralised analogies in his mind; and the most appropriate of these instantly suggests itself and leads him to a judicious arrangement of his troops.

[After a paragraph giving examples of several different sorts of practical skill based on past experience unaccompanied by any general rules, Mill continues:] Almost every one knows Lord Mansfield’s advice to a man of practical good sense who had been appointed governor of a colony and had to preside in its courts of justice, without previous judicial practice or legal education:

Give your decision boldly, for it will probably be right; but never give reasons for it, for they will almost certainly be wrong.

In cases like this (they are quite common) it would be absurd to suppose that the bad reason was the source of the good decision. Lord Mansfield knew that if any reason were given it would necessarily be an afterthought; the judge is guided by impressions from past experience, without taking the roundabout route through general principles based on them, and if he tries to construct any such principles he will assuredly fail. But Lord Mansfield wouldn’t have doubted that a man of equal experience who also had a mind stored with general propositions derived by legitimate induction from that experience would be greatly preferable as a judge to one who couldn’t be trusted to explain and justify his judgments, however wise they were. When talented men do wonderful things without knowing how, these are examples of the roughest and most spontaneous form of the operations of superior minds. It’s a defect in such men, and often a source of errors, not to have generalised as they went on; but generalisation, although it is a help—the most important of all helps—isn’t an essential.

Even scientifically educated people who have, in the form of general propositions, a systematic record of the results of the experience of all mankind, needn’t always bring in those general propositions when applying that experience to a new case. Dugald Stewart rightly says that although the reasonings in mathematics depend entirely on the axioms, we can see that a proof is conclusive without explicitly bringing in the axioms. The inference that AB is equal to CD because each of them is equal to EF will be accepted by everybody, including those who have never heard of the general truth that things that are equal to the same thing are equal to one another. When this remark of Stewart’s is consistently followed out, I think it goes to the root of the philosophy of ratiocination; and it’s unfortunate that he himself stopped short at a much more limited application of it. [Mill explains that Stewart thought he had a good point about axioms only, whereas really it holds for general propositions of all kinds. He continues:] This thoughtful and elegant writer has perceived an important truth, but only by halves. Having found that in the case of geometrical axioms general names have no magic power to conjure up new truths out of the well where they lie hidden, and not seeing that this is equally true of every kind of generalisation, he contended that axioms are barren of consequences and that the really fruitful truths—the real first principles of geometry—are the definitions. . . . Yet everything he had said about the limited function of the axioms in geometrical demonstrations is equally true of •the definitions. Every demonstration in Euclid could be carried on without •them. You can see this from the ordinary business of proving a geometrical proposition of by means of a diagram. What are our premises when we set out to demonstrate by a diagram some properties of the circle? Not that in all circles the radii are equal, but only that in this circle ABC the radii are equal. It’s true that to justify this assumption we appeal to the definition of circle in general; but all we need if for the assumption to be granted in the case of this particular circle ABC. From this singular proposition, combined with other propositions of a similar kind and other axioms, we prove that a certain conclusion is true not of all circles but of this particular circle ABC; or at least that the conclusion is true of this circle if our assumptions square with the facts. The. . . .general theorem that stands at the head of the demonstration is not the proposition that is actually demonstrated. Only one instance of it is demonstrated; but when we consider how this was done we see that the demonstration could be exactly copied in indefinitely many other instances—in every instance that conforms to certain conditions. The device of general language provides us with terms that connote these conditions, which lets us assert this indefinite multitude of truths in a single expression, and this expression is the general theorem. By dropping diagrams and replacing ‘ABC’ etc. by general phrases, we could prove the general theorem directly, i.e. demonstrate all the cases at once (of course having as our premises the axioms and definitions in their general form). But this only means that if we can prove an individual conclusion by assuming an individual fact, then whenever we are entitled to make an exactly similar assumption we can draw an exactly similar conclusion. The definition is a sort of notice to ourselves and others of what assumptions we think we’re entitled to make. . . .

§4. From the points I have been making the following conclusions seem to be established. All inference is from particulars to particulars: general propositions are merely registers of such inferences already made, and short formulae for making more. The major premise of a syllogism, therefore, is a formula of that sort, and the conclusion is not •an inference drawn from the formula but •an inference drawn according to the formula, because the real premise is the particular facts from which the general proposition was collected by induction. Those facts, and the individual instances that supplied them, may have been forgotten, but a record of them remains. Of course it isn’t a record that describes the facts themselves; but it shows what marks off the cases regarding which the facts (when they were known) were regarded as justifying a given inference. Guided by this record we draw our conclusion: which is in effect a conclusion from the forgotten facts! This requires us to read the record correctly, and the rules of the syllogism are a set of precautions to ensure our doing so.

This view of the functions of the syllogism is confirmed by consideration of the very cases that might be expected to be least favourable to it, namely those where ratiocination is independent of any previous induction. I have remarked that an ordinary syllogism is only the second half of the journey from premises to conclusion; but there are special cases in which it is the whole journey. All we can observe are particulars, so all knowledge derived from observation must begin with particulars; but in cases of certain descriptions our knowledge can be thought of as coming to us from sources other than observation. (a) It may present itself as coming from testimony, which in the given case is accepted as authoritative; and the information communicated by the testimony may involve not only particular facts but general propositions. That’s what happens when a scientific doctrine is accepted on the authority of writers, or a theological doctrine is accepted on the authority of Scripture. (b) Sometimes the generalisation isn’t an assertion (in the ordinary sense of the word) at all, but •a command, a •law in the moral and political sense of that word—an expression of a superior’s desire that we, or any number of other persons, shall act in accordance with certain general instructions. The fact which this asserts, namely a volition of the legislator, is an individual one; so the proposition is not a general proposition. But it contains a general description of the conduct the legislator wants his subjects to perform. The proposition asserts not that all men are anything, but that all men are to do something.

[The next two paragraphs use ‘authority’ and ‘witness’ in ways that might be found confusing but can be understood. Mill’s topic in (a) is testimony considered as providing the premise of an argument; in the context he is discussing the testimony is accepted, taken as •authoritative, and the testifier is a •‘witness’ in the now outdated sense of someone who asserts or assures us of something. The notion of a witness to particular events isn’t at work here, though it is in some other places in this work.]

In both these cases the generalities are the original data, and the particulars are derived from them by a process that is correctly represented as a series of syllogisms. But the real nature of the supposed deductive process is obvious enough. The only question is

(a) Did the authority who declared the general proposition intend to include this case in it? Or

(b) Did the legislator intend his command to apply to the present case?

This is answered by examining how the present case relates to what the authority or legislator had in mind. The object of the inquiry is to discover the witness’s or the legislator’s intention, through the indication given by their words. . . . The operation is a process not of inference but of interpretation.

That last phrase appears to me to characterise, more aptly than any other, the functions of the syllogism in all cases. When the premises are given by authority, the function of reasoning is to ascertain the testimony of a witness, or the will of a legislator, by interpreting the signs by which one has expressed his assertion and the other his command. Similarly, when the premises are derived from observation, the function of reasoning is to ascertain what we (or our predecessors) formerly thought might be inferred from the observed facts, and to do this by interpreting a memorandum of ours, or of theirs. The memorandum reminds us that evidence (more or less carefully weighed) led us to think that a certain attribute could be inferred wherever we perceive a certain mark. The proposition All men are mortal (for instance) shows that we have had experience from which we thought it followed that the attributes connoted by the term ‘man’ are a mark of mortality. But when we conclude that the Duke of Wellington is mortal, we infer this not from the memorandum but from the former experience. . . .

This view of the nature of the syllogism makes sense of something that is otherwise obscure and confused in the theory of Whately and other defenders of the syllogism regarding the limits to its usefulness. They say absolutely explicitly that the only role of general reasoning is to prevent inconsistency in our opinions—i.e. to prevent us from assenting to anything that would contradict something we had previously (on good grounds) assented to. And they say that the only reason a syllogism gives for assenting to the truth of the conclusion is that the supposition that it is false, combined with the supposition that both the premises are true, would lead to a contradiction. This would be a lame account of the real grounds we have for believing the facts that we learn from •reasoning as against •observation. The real reason why we believe that the Duke of Wellington will die is that his fathers and our fathers and all their contemporaries have died. Those facts are the real premises of the reasoning. But what leads us to infer the conclusion from those premises isn’t a need to avoid inconsistency! There’s no contradiction in supposing that all those persons have died and that the Duke of Wellington will live forever. But there would be a contradiction if we •were led by those same premises to make a general assertion including the case of the Duke of Wellington, and then •refused to stand by it in the individual case. We do have to avoid an inconsistency between •the memorandum we make of the inferences that can be justly drawn in future cases and •the inferences we actually draw in those cases. Just as a judge interprets a law so as to avoid giving any decision that doesn’t conform to the legislator’s intention, so also we interpret our own formula so as to avoid drawing inferences that don’t conform to our former intention. The rules for this interpretation are the rules of the syllogism: and its sole purpose is to maintain consistency between •the conclusions we draw in every particular case and •the previous general directions for drawing them—whether those general directions were formed by ourselves as the result of induction, or given to us by some competent authority.

§5. I think I have shown that although there’s always a process of reasoning or inference where a syllogism is used, the syllogism is not a correct analysis of that process, which in fact is an inference from particulars to particulars (except when it is a mere inference from testimony). This inference is authorized by a previous inference from particular propositions to general ones, and is substantially the same as that; so the inference is an instance of induction. But while these conclusions seem to me undeniable, I protest as strongly as Whately does against the doctrine that the syllogistic art is useless for the purposes of reasoning. The reasoning lies in the •act of generalisation, not in •interpreting the record of that act; but the syllogistic form runs an indispensable check on the correctness of the generalisation itself. [The point of that last clause: in arriving at ‘Bats have kidneys’ from ‘Bats are mammals’ and ‘All mammals have kidneys’ we are implicitly running a check on whether we were right to infer ‘All mammals have kidneys’ from the particular data from which we inferred it.]

We’ve seen that when we have a collection of particular propositions sufficient for grounding an induction we don’t have to form a general proposition; we may instead reason immediately from those particulars to other particulars. But if we’re entitled to infer a new •particular proposition, we are also entitled to infer a •general one. If from observation and experiment we can conclude to one new case, we can conclude to indefinitely many cases. . . . Every induction that suffices to prove one fact proves an indefinite multitude of facts; the experience that justifies a single prediction must suffice to support a general theorem. It’s extremely important to discover this theorem and state it in a general a form as possible declare; in that way we place before our minds the whole range of what our evidence must prove if it proves anything.

. . . .In reasoning from a set of individual observations to some new and unobserved case that we’re not perfectly acquainted with (or we wouldn’t be inquiring into it), and that we are probably especially interested in (if not, why are we inquiring into it?), there’s very little to protect us from •becoming careless, or from •letting our thought be biased by our wishes or our imagination and thus •accepting insufficient evidence as sufficient. But if instead of concluding straight to the particular case x we place before ourselves an entire class of facts—a general proposition every bit of which is legitimately inferable from our premises if the inference to x is legitimate—then it’s quite likely that if the premises don’t support the generalisation it will contain within it some factual proposition which we already know to be false; in that way we’ll discover the error in our generalisation ·and be led by that to back off from the inference to x·.

Consider a Roman citizen during the reign of Marcus Aurelius who expects that the emperor’s son Commodus will be a just ruler; he is led to that by a natural bias in his thinking produced by the ·excellent· lives and characters of the last few emperors and the present one; and he has a nasty shock when Commodus becomes emperor. He might be saved from this by reflecting that his expectation regarding Commodus couldn’t be justifiable unless his evidence for it also entitled him to infer some general proposition, e.g. that all Roman emperors are just rulers; that would immediately have reminded him of Nero, Domitian, and other bad emperors, showing him •the falsity of the general proposition and therefore •the insufficiency of his evidence for it, and thus the insufficiency of that same evidence to prove the favourable proposition about Commodus.

Everyone agrees that when an inference is challenged, it’s a help to consider parallel cases. Well, by ascending to the general proposition we bring into view not a mere one parallel case but all possible parallel cases—all cases to which the same set of evidentiary considerations are applicable.

·Summing up·: When we argue from a number of known cases to another case that we think is analogous, it is always possible and usually worthwhile to take our argument by the longer route through •an induction from those known cases to a general proposition followed by •an application of that general proposition to the unknown case. This second part of the operation, which is essentially a process of interpretation, will come down to a syllogism or a series of syllogisms in which the major premises will be general propositions covering whole classes of cases; and every one of these must be true across its whole range if the argument is maintainable. Thus, if any fact in the range of one of these general propositions—and consequently asserted by it—is known or suspected to be other than the proposition asserts it to be, this leads us to know or suspect that the original observations that are the real grounds of our conclusion are not sufficient to support it. And the greater chance of our detecting the weakness of our evidence, the more confident we are entitled to have ·in our conclusion· if no weakness in the evidence appears.

So the value of the syllogistic form and of the rules for using it correctly does not consist in their

being the form and the rules according to which our reasonings must be made or even usually are made;

but in their

•providing us with a way of formulating those reasonings that is admirably fitted to bring their inconclusiveness to light if they are inconclusive.

An induction from particular propositions to general ones, followed by a syllogistic process from the latter to other particulars. . . ., is a form in which we may reason whenever we choose, and must adopt when there’s doubt as to whether we are reasoning validly; though when the case is familiar and not very complicated, and there’s no suspicion of error, we may and do reason immediately from the known particular cases to unknown ones.[2]

The further uses of the syllogism in the general course of our intellectual operations hardly require illustration, because they are just the known uses of general language. They amount to this:

The inductions can be made once for all: a single careful interrogation of experience may be enough; and the result can be registered in the form of a general proposition that is committed to memory or to writing, and can then be put to work in syllogisms. The details of our experiments can then be dismissed from the memory (which couldn’t retain them all); while the knowledge those details provided for future use. . . .is retained in a commodious and immediately available shape by means of general language.

The down-side of this advantage is that inferences made on insufficient evidence become consecrated—hardened (so to speak) into general maxims—and the mind clings to them from habit, after it has outgrown any liability to be misled by similar appearances if they were now presented for the first time. Because it has forgotten the details, it doesn’t think of revising its own former decision. This is an inevitable •drawback, and not a trivial one; but it is greatly outweighed by the immense •benefits of general language.

The use of the syllogism is simply the use of general propositions in reasoning. We can reason without them; in simple and obvious cases we do; minds of great sagacity can do it in cases that aren’t simple and obvious, provided their experience has provided them with instances essentially similar to every combination of circumstances likely to arise. But lesser minds, and the same minds when not so well supplied with relevant personal experience, are helpless without the aid of general propositions, wherever the case presents the smallest complication. If we didn’t make general propositions, few of us would get much further than the simple inferences drawn by the more intelligent of the lower animals. Though not necessary to reasoning, general propositions are necessary to any considerable progress in reasoning. So it’s natural—indeed indispensable—to split the process of investigation into two parts: (i) obtain general formulae for determining what inferences may be drawn, then (ii) draw the inferences. In drawing them we are applying the formulae; and the rules of syllogism are a system of securities for the correctness of the application.

§6. Given that the syllogism is not the universal type [see Glossary] of the reasoning process, what is the real type? This comes down to the question:

What is the nature of the minor premise, and how does it contribute to establishing the conclusion?

We now fully understand that the place that the major premise nominally occupies in our reasonings really belongs to the individual facts of which it expresses the general result. It isn’t a real part of the argument, but only an intermediate halting-place for the mind, inserted between the real premises and the conclusion as an important safeguard of the correctness of the process. But the minor premise is an indispensable part of the syllogistic expression of an argument: it certainly •is or •corresponds to an equally indispensable part of the argument itself, and we have only to inquire what part. [Note the distinction between ‘part of the expression of the argument’ and ‘part of the argument itself’.]

Thomas Brown. . . .saw the begging of the question that is inherent in every syllogism if we ·wrongly· consider the major premise to be itself the evidence by which the conclusion is proved; he didn’t see the immense advantage in security-for-correctness that we get from interposing this step between the real evidence and the conclusion; but he entirely deleted the major premise from the reasoning process, without putting anything in its place. He maintained that our reasonings consist only of the minor premise and the conclusion: Socrates is a man, therefore Socrates is mortal, thus suppressing the appeal to former experience as an unnecessary step in the argument! He didn’t see the absurdity of this because of his opinion that •reasoning is merely •analysing our own general notions or abstract ideas, and that the proposition Socrates is mortal is evolved from the proposition Socrates is a man simply by recognising the notion of mortality as already contained in our notion of man.

[Mill devotes a long paragraph to pursuing Brown. The central point is that Brown sees the need for something to connect ‘Socrates is a man’ with ‘Socrates is mortal’; tries to supply it with a thesis about connections between ‘ideas’; and Mill argues convincingly that this move either fails utterly or is a reworded version of the generalisation, the erstwhile ‘major premise’ that Brown has banished. Now, with Brown moved out of the way, Mill returns to the question of the ‘universal type’ of the reasoning process:]

In the argument proving that Socrates is mortal, one indispensable part of the premises will be this:

‘My father and his father and A and B and C and indefinitely many other persons, were mortal’

which is merely one way of saying that they died. This is the major premise, no longer begging the question and cut down to what is really known by direct evidence.

To link this proposition with the conclusion ‘Socrates is mortal’ what is needed is the proposition:

‘Socrates resembles my father and his father and A and B and C and all the other individuals specified.’

That’s what we assert when we say that Socrates is a man. By saying this we also assert in what respect he resembles them, namely in the attributes connoted by ‘man’. And we conclude that he further resembles them in the attribute mortality.

§7. So now we have what we were looking for, a universal type of the reasoning process, which comes down to this: Certain individuals have a given attribute; one or more individuals resemble the former in certain other attributes; therefore they resemble them also in the given attribute. This doesn’t claim, as the syllogism does, to be conclusive merely because of its form; it can’t possibly be so. That Q does or doesn’t assert the fact that was already asserted in P may be shown by the forms of the expressions, i.e. by a comparison of P’s wording with Q’s; but if they assert facts that are genuinely different, the question of whether P’s truth proves Q’s can’t be shown by •the language they’re expressed in, but must depend on •other considerations. Given the attributes in which Socrates resembles the men who have already died, is it permissible to infer that he resembles them also in being mortal? That is a question of induction, and is to be decided by the principles or canons which test the correct performance of that great mental operation. I’ll come to those in due course.

If an inference can be drawn regarding •Socrates then it can be drawn regarding •everyone who resembles the observed individuals in the same way that he resembles them, i.e. regarding all mankind. If the argument is admissible in the case of Socrates, therefore, we’re free once for all to treat the possession of the attributes of man as a mark, i.e. as satisfactory evidence, of the attribute of mortality. We do this by asserting the universal proposition All men are mortal and interpreting this in its application to Socrates and to others as occasion arises. This conveniently divides the entire logical operation into two steps:

(1) ascertaining what attributes are marks of mortality;

(2) ascertaining whether any given individuals possess those marks.

And when we are theorising about the reasoning process, it will generally be advisable to regard this double operation as actually happening, pretending that all reasoning is carried on in the form that it has to be given if we are to subject it to any test of its correct performance.

Every process of thought in which the basic premises are particular propositions—whether the conclusion is a general formula or other particular propositions—is a case of induction; but I’ll stay in line with ordinary usage by reserving the name ‘induction’ for the process of establishing the general proposition, and I’ll give to the remaining operation—which is substantially the process of interpreting the general proposition—its usual name ‘deduction’. And I’ll regard every process in which something is inferred regarding an unobserved case as consisting of •an induction followed by •a deduction. The process doesn’t have to be carried out in this form, but it always can be, and it must be put into this form when assurance of scientific accuracy is needed and desired.

§8. The theory of the syllogism I have been presenting has been accepted by some important thinkers, three of whom are especially valuable allies: Sir John Herschel, Dr. Whewell, and Mr. Bailey. Of these, Herschel regards the doctrine as ‘one of the greatest steps that have yet been made in the philosophy of logic.’[3] ‘When we consider’ (quoting Herschel) ‘the deeply ingrained status of the habits and prejudices that it has cast to the winds’, there’s no cause for anxiety in the fact that other equally respectable thinkers have formed a very different estimate of it. Their principal objection is compactly stated in a sentence by Whately:

‘In every case where an inference is drawn from induction (unless the name ‘induction’ is to be given to a mere random guess without any grounds at all) we must form a judgment •that the instance or instances adduced are sufficient to authorise the conclusion; •that it is allowable to take these instances as a sample warranting an inference regarding the whole class’

and the expression of this judgment in words (it has been said by several of my critics) is the major premise.

The major ·premise· affirms the sufficiency of the evidence on which the conclusion rests—I don’t just admit this; it’s the essence of my own theory. Anyone who admits that the major premise is only this adopts my theory in its essentials.

But this recognition of the sufficiency of the evidence—i.e. of the correctness of the induction—is not a part of the induction itself. (If it is, we’ll have to accept that a part of everything we do is to satisfy ourselves that we’ve done it rightly!) We conclude from known instances to unknown by the impulse of our generalising propensity; it’s only after much practice and mental discipline that we raise the question of the sufficiency of the evidence; and when we raise it we go back along our path and examine whether we were justified in doing what we have provisionally done. To speak of this reflex operation as part of the original one, having to be expressed in words so that the verbal formula will depict the psychological process, strikes me as false psychology.[4] We review our syllogistic processes as well as our inductive ones, and recognise that they have been correctly performed; but logicians don’t add a third premise to the syllogism to express this act of recognition. A careful copyist verifies his transcript by collating it with the original;. . . .but we don’t call this process a part of the act of copying!

The conclusion in an induction is inferred from •the evidence itself, not from •a recognition of the sufficiency of the evidence. I infer that my friend is walking towards me because •I see him, and not because •I recognise that my eyes are open and that eyesight is a means of knowledge. In all operations that require care it’s good to assure ourselves that the process has been performed accurately; but the testing of the process is not the process; and even if testing is omitted altogether, the process may still be correct. It’s just because the testing is omitted in ordinary unscientific reasoning that anything is gained in certainty by putting reasoning into the syllogistic form. Doing our best to make sure that it isn’t omitted, we make the testing operation a part of the reasoning process itself: we insist that the inference from particulars to particulars shall pass through a general proposition. But this is •a security for good reasoning, not •a condition of all reasoning; and in some cases it isn’t even a security. All our most familiar inferences are made before we learn the use of general propositions; and a person with high untrained intelligence will skillfully apply his acquired experience to adjacent cases, though he would bungle grievously if he tried to fix the limits of the appropriate general theorem. But though he may conclude rightly, he doesn’t strictly know whether he has done so, because he hasn’t tested his reasoning. That is exactly what forms of reasoning do for us. We need them to enable us not to reason but to know whether we reason correctly.

And here’s another point against the objection. Even when the test has been applied and the sufficiency of the evidence recognised, if it’s sufficient to support the general proposition it is also sufficient to support an inference from particulars to particulars without passing through the general proposition. . . . The general conclusion is never legitimate unless the particular one would be so too; and in no intelligible sense can the particular conclusion be said to be ‘drawn from’ the general one. Whenever there is ground for drawing •any conclusion from particular instances there’s ground for •a general conclusion; but however useful it may be to actually draw this conclusion, this can’t be required for the validity of the inference in the particular case. . . .

[Mill ends this section with a long footnote replying to an unnamed reviewer of an earlier edition of this work. It’s interesting, but doesn’t add much to what has been said in the main text. One lordly put-down is memorable: ‘If the reviewer does not see that there is a difficulty here, I can only advise him to reconsider the subject until he does: after which he will be a better judge of the success or failure of an attempt to remove the difficulty.’]

§9. These considerations enable us to understand •the true nature of what recent writers have called ‘formal logic’, and •the relation between it and logic in the widest sense of that term. Logic as I conceive it is the entire theory of the ascertainment of reasoned or inferred truth. So formal logic—which Hamilton and Whately have both, from their different points of view, represented as the whole of logic properly so-called—is really a very subordinate part of it, because it’s not directly concerned with reasoning or inference in the sense in which that process is a part of the investigation of truth. What, then, is formal logic? The name seems to be properly applied to all the doctrine relating to the equivalence of different modes of expression; the rules for determining when assertions in a given form imply or presuppose the truth or falsity of other assertions. This includes the theory

•of the import of propositions, and of their conversion, equivalence and opposition;

•of the wrongly so-called ‘inductions’ where the ‘generalisation’ is a mere abridged statement of cases known individually (I’ll discuss these in III.2.2); and

•of the syllogism.

The theory of Naming, and of (what is inseparably connected with it) Definition, though belonging more to the other and larger kind of logic than to formal logic, is a necessary preliminary to the latter also. The end aimed at by formal logic, and attained by obeying its rules, is not truth but consistency. I have shown that this is the only direct purpose of the rules of the syllogism; their intention and effect is simply to keep our inferences or conclusions consistent with our general formulae or directions for drawing them. The logic of consistency is a necessary auxiliary to the logic of truth, for two reasons. (i) What is inconsistent with itself or with other propositions that are true can’t itself be true. (ii) Truth can be successfully pursued only by drawing inferences from experience; if these are justifiable at all they can be generalised, and for their justification to be tested they have to be stated in a generalised form; after which the correctness of their application to particular cases is a question that specially concerns the logic of consistency. This logic doesn’t require any previous knowledge of the processes or conclusions of the various sciences, so it can profitably be studied at a much earlier stage of education than can the logic of truth. . . .

NOTES

[1] Of course I am not defending any such absurdity as that we actually ‘ought to have known’ and considered the case of every individual man, past, present, and future, before affirming that all men are mortal: although this interpretation has been put upon what I have been saying. I don’t differ from Whately, or from any other defender of the syllogism, on the practical part of the matter; I’m only pointing out an inconsistency in the logical theory of the syllogism as conceived by almost all writers. I don’t say that a person who before the Duke of Wellington was born affirmed that all men are mortal knew that the Duke of Wellington was mortal; but I do say that he asserted it; and I ask for an explanation of the apparent logical fallacy of adducing in proof of the Duke of Wellington’s mortality, a general statement that presupposes it. Finding no good solution of this problem in any of the writers on logic, I have tried to provide one.

[2] The language of ratiocination would fit the real nature of the process better if the relevant general propositions were expressed not in the form ‘All men are mortal’ or ‘Every man is mortal’ but rather in the form ‘Any man is mortal’. Then we would have the likes of ‘The men A, B, C etc. are thus-and-so, therefore any man is thus-and-so’—which is a better representation of the truth that •inductive reasoning is always basically inference from particulars to particulars, and that •the only role of general propositions in reasoning is to vouch for the legitimacy of such inferences.

[3] He says that it’s not strictly ‘a discovery’ because Berkeley got there first. In a recent careful re-reading of Berkeley’s whole works, I haven’t found this doctrine in them. Herschel probably meant that it’s implied in Berkeley’s argument against abstract ideas. But I can’t find that Berkeley saw the implication, or ever asked himself what bearing his argument had on the theory of the syllogism. . . .

[4] See the important chapter on Belief in Bain’s great treatise The Emotions and the Will.

Chapter 4. Trains of reasoning, and deductive sciences

§1. In my analysis of the syllogism, we saw that the •minor premise always affirms a resemblance between a new case and some cases previously known; while the •major premise asserts something which, having been found true of those known cases, we think we’re entitled to hold true of any other case resembling the former in certain given respects.

In each example presented in chapter 3, the minor premise asserts a resemblance that is obvious to the senses, such as ‘Socrates is a man’. If every ratiocination had a minor premise like that, there would be no need for trains of reasoning, and deductive or ratiocinative sciences [see Glossary] wouldn’t exist. Trains of reasoning exist only for the sake of extending an induction that is based (as all inductions must be) on observed cases to other cases in which we can’t directly observe the fact which is to be proved and can’t even directly observe the mark that is to prove it.

§2. Consider the syllogism:

•All cows ruminate,

•This animal right here a cow; therefore

•This animal ruminates.

If the minor premise is true it is obviously so; it’s only the major premise that has to be established through a previous process of inquiry; and provided the induction that the major premise expresses was correctly performed, the conclusion about the present animal will be instantly drawn because as soon as she is compared with the formula she will be identified as being included in it. But now consider this:

•All arsenic is poisonous;

•This substance right here is arsenic; therefore

•This substance is poisonous.

The truth of this minor premise may not be obvious at first sight; it may be known only by inference as the conclusion of another argument which, put into the syllogistic form, goes like this:

•Anything which when lighted produces a dark spot on a piece of white porcelain held in the flame, the spot being soluble in hypochloride of calcium, is arsenic;

•This substance right here conforms to this condition; therefore

•This substance is arsenic.

Thus, to establish the final conclusion that this substance is poisonous we need a process which. . . .stands in need of two syllogisms; and we have a train of reasoning.

But when in this way we add syllogism to syllogism, we’re really adding induction to induction. For this chain of inference to be possible there must have been two separate inductions. They may well have been based on different sets of individual instances, but they’ll have converged in their results so that the instance that is now the subject of inquiry comes within the range of them both. The record of these inductions is contained in the major premises of the two syllogisms. First observation: we or others have examined various objects which under the given circumstances yielded a dark spot with the given property, and found that they had the properties connoted by ‘arsenic’—they were metallic, volatile, their vapour had a smell of garlic, and so on. Second observation: We or others have examined various specimens that had this metallic and volatile character, whose vapour had this smell, etc., and have found them all to be poisonous. First induction: We judge that we may extend the first observation to all substances yielding that particular kind of dark spot. Second induction: We judge that we may extend the second observation to all metallic and volatile substances resembling those we examined; and consequently not only to those that have •been seen to be such, but also to those that are •concluded to be such by the first induction. The substance before us is only seen to come within the scope of the first induction; but by means of this it is brought within the scope of the second. We are still concluding from particulars to particulars; but now we are concluding from observed particulars to other particulars that aren’t—as in the simple ·one-syllogism· case—•seen to resemble them in the relevant respects but are •inferred to do so because they resemble them in something that we have been led by quite a different set of instances to consider as a mark of the former resemblance.

This first example of a train of reasoning is extremely simple—a series consisting of only two syllogisms. Here’s a somewhat more complicated example:

•No government that earnestly seeks the good of its subjects is likely to be overthrown;

•The government of X earnestly seeks the good of its subjects; therefore

•The government of X is not likely to be overthrown.

I’m supposing that the major premise is not derived from considerations a priori but is a generalisation from history. So it was based on observation of governments concerning whose desire for the good of their subjects there was no doubt. It has been found, or thought to be found, that these governments were not easily overthrown, and it has been judged that those instances justified an extension of the same predicate (·not easily overthrown·) to every government that resembles them in the attribute of desiring earnestly the good of its subjects. But does the government of X resemble them in this respect? This. . . .would have to be proved by another induction, for we can’t directly observe the sentiments and desires of the members of the government of X. To prove the minor premise, therefore, we need an argument in this form:

•Every government that acts in manner M desires the good of its subjects;

•The government of X acts in manner M: therefore

•The government of X desires the good of its subjects.

But is it true that the government acts in manner M? This minor also may require proof, by still another induction:

•Whatever is asserted by intelligent and disinterested witnesses may be believed to be true;

•That the government of X acts in manner M is asserted by intelligent and disinterested witnesses; therefore

•That the government of X acts in manner M may be believed to be true.

So the argument consists of three steps. Having the evidence of our senses that the case of the government of X resembles a number of former cases in having something said about it by intelligent and disinterested witnesses, we infer (i) that as in those former instances so also in this one, the assertion is true. The assertion in question was that the government of X acts in manner M; other governments or persons have been observed to act in that manner, and they are known to have desired the good of the people; and we infer (ii) that the government of X resembles those others not only in its manner of governing but also in desiring the good of its people. This brings the government of X into known resemblance with the other governments that were thought likely to escape revolution; and so by a third induction we infer (iii) that the government of X is also likely to escape. This is still reasoning from particulars to particulars, but here we are reasoning to the new instance from three distinct sets of former instances. With only one of these sets of instances—

governments that have been said by intelligent and disinterested witnesses to act in manner M

—do we directly perceive the government of X to be similar; from that similarity we inductively infer that it has the attribute which makes it resemble the second set—

governments that act in manner M

—and that resemblance is our basis for a further induction through which we assimilate the government of X with a third set of instances—

governments that desire the good of their subjects

—and from there we perform our final induction, bringing the government of X into the class of

governments that are not likely to be overthrown

which gives us the final conclusion. [Mill rightly says ‘three sets of instances’; the fourth set he has mentioned is not something we are reasoning from.]

§3. Everything that I said in chapter 3 about the general theory of reasoning holds just as much for these more complex examples as it did for chapter 3’s simpler ones. The successive general propositions are not steps in the reasoning; they aren’t intermediate links in the chain of inference between the observed particulars and the conclusions we draw from them. If we had big enough memories, and enough power to maintain order among a huge mass of details, the reasoning could go through without any general propositions; they are mere formulae for inferring particulars from particulars. [Mill now repeats his thesis about the role of general propositions in reasoning, summing up thus:] The real inference is always from particulars to particulars, from the observed instances to an unobserved one: but in drawing this inference we conform to a formula that we have adopted for our guidance in such operations; it’s a record of the criteria by which we thought we could draw the line between legitimate and illegitimate inferences. The real premises are the individual observations. We may have forgotten them (or indeed have never known them, because they weren’t made by us). But we have before us ·the general proposition, which provides· proof that we or others once thought those observations to be sufficient for an induction; and any new case has marks tell us whether it would have fallen within the scope of the original induction if it had been known at that time. We may recognise these marks at once, or we may recognise them through the aid of other marks which we take to be marks of the first, on the strength of a previous induction. It may be that these marks of marks are recognised only through a third set of marks;. . . and so on. We can have a train of reasoning of any length to bring a new case within the scope of an induction based on particulars whose similarity to the new case is ascertained only in this indirect manner.

Thus, in the preceding example, the final inductive inference was that the government of X was not likely to be overthrown; this inference was drawn according to a formula in which desire for the public good was set down as a mark of not being likely to be overthrown; a mark of this mark was acting in manner M; and a mark of acting in manner M was being asserted to do so by intelligent and disinterested witnesses: and our senses told us that the government of X possessed this mark. Hence that government fell within the last induction, which brought it within all the others. The perceived resemblance of the case to one set of observed particular cases brought it into known resemblance with another set, and that with a third.

[In this paragraph ‘→’ replaces Mill’s ‘a mark of’.] In the more complex branches of knowledge, the deductions seldom consist of a single chain—

a → b

b → c

c → d, therefore

a → d.

They consist (to carry on the same metaphor) of several chains united at the end-point, like this, for example:

a → d

b → e

c → f

d e f → n, therefore

a b c → n.

[Mill gives an example of a complex inference in optics that has exactly that form, and comments:] Most chains of physical deduction are of this more complicated type; and they occur frequently in mathematics, e.g. in all propositions where the hypothesis includes numerous conditions: ‘If a circle be taken, and if within that circle a point be taken, not the centre, and if straight lines be drawn from that point to the circumference, then. . . ’ etc.

§4. The view I have taken of reasoning might seem hard to reconcile with the fact that there are deductive or ratiocinative sciences. This might be said:

‘If all reasoning is induction, all the difficulties of philosophical investigation must lie in the inductions; and when these are easy and not open to doubt or hesitation, there could be no science, or anyway no difficulties in science. For example, the existence of an extensive science of mathematics, requiring the highest scientific genius in those who contributed to its creation, and calling for a most continued and vigorous exertion of intellect in order to appropriate it when created, is hard to account for on Mill’s theory.’

But what I have been saying in this chapter enables me to remove this difficulty. I have shown that even when the inductions themselves are obvious, it may be really difficult to discover whether the particular case Q we are investigating comes within their scope; and there’s plenty of room for scientific ingenuity in combining various inductions in such a way that, by means of one that obviously has Q in its range, Q can be brought within the scope of others that aren’t obviously relevant to it.

When in a science the more obvious inductions from direct observations have been made, and general formulas have been framed setting the limits to the range of applicability of these inductions, if every new case that comes up can be at once seen to fall under one of the formulas, the induction is applied to the new case and the business is ended. But it often happens that a new case x arises that doesn’t obviously come within the range of any formula that could answer the question we’re asking about x. Let us take an instance from geometry.[1] My example will be the fifth proposition of the first book of Euclid. The question to be answered is: Are the angles at the base of an isosceles triangle equal or unequal? Well, what inductions [meaning: was inductively reached conclusions] do we have from which we can infer equality or inequality. For inferring equality:

•Things that coincide when they are applied to each other are equals.

•Things that are equal to the same thing are equals.

•A whole and the sum of its parts are equals.

•The sums of equal things are equals.

•The differences of equal things are equals.

There are no other basic formulae to prove equality. For inferring inequality:

•A whole and its parts are unequals.

•The sums of equal things and unequal things are unequals.

•The differences of equal things and unequal things are unequals.

In all, eight formulae. The angles at the base of an isosceles triangle don’t obviously come within the range of any of these. The formulae specify certain marks of equality/inequality, but the angles can’t be perceived intuitively to have any of those marks. But on examination it appears that they have; and we eventually succeed in bringing them under the formula ‘The differences of equal things are equal’. Why is it hard to recognise these angles as the differences of equal things? Because each of them is the difference not of merely one pair but of innumerable pairs of angles; and we had to imagine and select two that could either •be intuitively perceived to be equals or •had some of the marks of equality set down in the various formulae. By an exercise of ingenuity (the first time it was done it was a considerable exercise of ingenuity) two pairs of angles were identified that united these requisites: (1) That their differences were the angles at the base ·of an isosceles triangle· could be perceived intuitively; and (2) they had one of the marks of equality, namely coincidence when applied to one another. This coincidence wasn’t perceived intuitively, but inferred in conformity with another formula.

For greater clearness, I offer an analysis of the demonstration. Euclid demonstrates his fifth proposition by means of the fourth; but I can’t do that because I have undertaken to trace deductive truths not to •prior deductions but to •their original inductive foundation. So I must use the premises of the fourth proposition instead of its conclusion, and prove the fifth directly from first principles. This requires six formulas. [Mill does a conscientious job of proving the proposition and showing how each step in the proof fits with his theory of deduction. It makes for laborious reading, though, and we can skip it at this stage without harming our ability to what follows. The proof is given at the end of this chapter here.]

The main problem here was to see the two angles at the base of the triangle ABC as remainders made by cutting one pair of angles out of another, while the members of each pair are corresponding angles of triangles that have two sides and the intervening angle equal. It’s this happy contrivance that brings so many different inductions to bear on this one particular case. And because this far from obvious procedure has a role to play so near to the threshold of mathematics, you can see how much scope there may be for scientific dexterity in the higher branches of that and other sciences, in order to combine a few simple inductions so as to bring within each of them countless cases that aren’t obviously included in it. And you can also see that the processes needed for bringing the inductions together in the right way may be long and complicated, even when each separate induction is easy and simple. All the inductions involved in all of geometry are comprised in those simple ones, the formulae of which are the axioms and a few of the so-called ‘definitions’. The remainder of the science is made up of the work of bringing unforeseen cases within these inductions; or (in syllogistic language) of proving the minor premises needed to complete the syllogisms—the major premises being the definitions and axioms. Those definitions and axioms present all the marks by a skillful combination of which it has been found possible to discover and prove everything that is proved in geometry. The marks are few in number, and the inductions that provide them are obvious and familiar; so the whole difficulty of geometry has to do with connecting several of them together so as to construct deductions, i.e. trains of reasoning. Doing this is the science of geometry. . . ., so geometry is a deductive science.

§5. In III.4.3 and elsewhere I’ll show that there are weighty scientific reasons for making every science a deductive science as far as possible. That is, we should try to construct the science from the fewest and the simplest possible inductions, and to make these—by any combinations, however complicated—enough to prove the science’s results. And all these, even the very complex results, could if we chose be proved by inductions from specific experience. Every branch of natural science was originally experimental; each generalisation rested on a special induction, and was derived from its own separate set of observations and experiments. From being so-called ‘sciences of pure experiment’—or, more correctly, sciences in which most of the reasonings involve only one step and are expressed by single syllogisms—all these sciences have become to some extent sciences of pure reasoning, in which many truths already known by induction from many different sets of experiments are exhibited as deductions or corollaries from inductive propositions of a simpler and more universal character. (I said ‘to some extent’; with some sciences it is nearly their whole extent.) Thus mechanics was made mathematical, then hydrostatics, then optics, then acoustics, then thermology; and Newton brought astronomy within the laws of general mechanics. . . . Although by this progressive transformation all sciences become increasingly •deductive, that doesn’t mean that they become less •inductive; every step in a deduction is an induction. The opposition is not between deductive and inductive, but between deductive and experimental. A science is experimental to the extent that every new case with new features needs a new set of observations and experiments—a fresh induction. It is deductive to the extent that it can deal with cases of a new kind by bringing them under old inductions, doing this by ascertaining that cases that can’t be observed to have the relevant marks do have marks of those marks.

[In this paragraph and the next, ‘a → b’ replaces Mill’s ‘a is a mark of b’, and ‘a ↔ b’ replaces his ‘a and b are marks of one another’.] So now we can see what the general distinction is between •sciences that can be made deductive and •sciences that must as yet remain experimental. It depends on whether we have been able to discover marks of marks. If our various inductions haven’t let us get any further than such propositions as

a → b or a ↔ b

c → d or c ↔ d

without anything to connect a or b with c or d, then we have a science of detached and mutually independent generalisations, such as

•‘Acids redden vegetable blues,

•Alkalis colour them green,

from neither of which propositions could we directly or indirectly infer the other; and to the extent that a science is composed of propositions like that, it is purely experimental. Chemistry, in the present state of our knowledge, hasn’t yet escaped from being like this. [An essential part of that escape was the discovery of the Periodic Table by Dmitri Mendeleev in 1869—26 years after the first edition of Mill’s System of Logic and 13 years before the eighth edition, which is what we are reading.] But there are other sciences containing propositions of this kind:

•a → b

•b → c

•c → d

•d → e

and so on. In these sciences we can climb the ladder from a to e by a process of ratiocination; we can conclude that a is a mark of e, and that every object that has the mark a has the property e; even if we have never been able to observe a and e together; and even if d, our only direct mark of e, isn’t •perceptible but only •inferable in the objects to which we attribute it. Or, moving from ‘chains’ to a different metaphor, we may be said to get from a to e underground: the marks b, c, d, which indicate the route must all be possessed somewhere by the objects we are investigating, but they are below the surface. The only visible mark is a, and by it we can trace in succession all the rest.

§6. We can now understand how an experimental science may become deductive science merely by the progress of experiment. In an experimental science, the inductions are detached—a → b, c → d, e → f, and so on—but at any time a new set of instances, and thus a new induction, may bridge the gap between two of these unconnected arches. For example., it may turn out that b → c, which enables us now to prove deductively that a → c. And it sometimes happens that some comprehensive induction raises an arch high in the air, bridging over hosts of gaps all at once, so that b, d, f and all the rest turn out to be marks of some one thing, or of different things between which a connection has already been traced. Newton discovered that all the motions of all the bodies in the solar system (each of which motions had been inferred by a separate logical operation, from separate marks) were marks of

moving around a common centre, with a centripetal force varying directly as the mass, and inversely as the square of the distance from that centre.

This is the greatest example that has yet occurred of a science that is to a large extent merely experimental being transformed in one stroke into a deductive science.

Transformations like that but on a smaller scale continually take place in the less advanced physical sciences without enabling them to escape the status of experimental sciences. Thus regarding the unconnected propositions cited in §5—

•Acids redden vegetable blues

•Alkalis make them green

—Liebig has found that there is nitrogen in all blue colouring matters that are reddened by acids as well as all red colouring matters that are turned blue by alkalis; and this fact may some day provide a connection between those two propositions, by showing that the antagonistic action of acids and alkalis in producing or destroying the colour blue is the result of some one more general law. Whenever detached generalisations come to be connected, that is something gained; but it doesn’t do much to give a deductive character to any science as a whole, because the observations and experiments that enable us to inter-connect a few general truths usually reveal to us a greater number of unconnected new ones. Generalisations in chemistry are continually being extended and simplified in this way, but chemistry is mainly an experimental science, and is likely to remain so unless some comprehensive induction is arrived at, which (like Newton’s) inter-connects a vast number of the smaller known inductions and immediately changes the whole method of the science. Chemistry has already one great generalisation, which possesses this comprehensive character within one part of chemistry: namely, Dalton’s principle—the ‘atomic theory’ or the doctrine of ‘chemical equivalents’—which enables us to a certain extent to know in advance the proportions in which two substances will combine. This is undoubtedly a source of new chemical truths obtainable by deduction, as well as a connecting principle for all similar truths that were previously obtained by experiment.

§7. The discoveries that change a science from experimental to deductive mostly consist in establishing (by deduction or by direct experiment) that the varieties of some familiar kind of phenomenon are uniformly accompanied by varieties of some other phenomenon. The science of sound, which previously stood in the lowest rank of merely experimental sciences, became deductive when experiments showed that every variety of sound was a result of, and therefore a mark of, a distinct and definable variety of wave motion among the particles of the transmitting medium. When this was ascertained, it followed that every relation of succession or co-existence which obtained between phenomena of the more familiar class (·sound·) obtained also between the corresponding phenomena in the other class (·wave-motion·). Every sound, being a mark of a particular wave-motion, became a mark of everything that could be inferred from that motion by the laws of dynamics; and everything that was (according to those same laws) a mark of any wave-motion among the particles of an elastic medium became a mark of the corresponding sound. In this way many previously unsuspected truths about sound can be deduced from the known laws of the propagation of motion through an elastic medium; while empirically known facts about sound come to indicate previously unknown properties of vibrating bodies.

But the grand agent for turning experimental sciences into deductive ones is the science of number. The properties of number are the only known phenomena that are in the strictest sense properties of all things whatsoever. It’s not the case that all things have colour, weight, or even size, but all things are numerable [= ‘can be counted’]. And if we consider this science in its whole extent, from common arithmetic up to the calculus of variations, there seem to be almost countless truths already known, with a promise of indefinitely more.

These truths apply to things only in respect of their •quantity. But if we discover that variations of •quality in some class of phenomena correspond regularly to variations of •quantity in those same phenomena or in some others, every mathematical formula that applies to quantities which vary in that particular manner becomes a mark of a corresponding general truth about the variations in quality that accompany them; and because the science of quantity is entirely deductive (as far as any science can be), the theory of that particular kind of qualities becomes to this extent deductive likewise.

The most striking example of this kind of transformation is the revolution in geometry that originated with Descartes and was completed by Clairaut. This didn’t involve an experimental science’s becoming deductive; it started with a science that was already deductive and increased—to an unparalleled extent—the range of its deductive processes. These great mathematicians pointed out the importance of the fact that to every variety of

•position of points,

•direction of lines, and

•shape of curves or surfaces

—all of which are •qualities—there is a corresponding relation of •quantity between either two or three straight-line co-ordinates. The upshot of this is that if we know the law according to which those co-ordinates vary relatively to one another, we can infer every other geometrical property— quantitative or qualitative—of the line or surface in question. From this it followed that every geometrical problem could be solved if the corresponding algebraic one could; and geometry received an accession (actual or potential) of new truths, corresponding to every property of numbers that the progress of the calculus had brought (or might in future bring) to light. Mechanics, astronomy, and (in a lesser degree) every branch of natural science have been made algebraic in the same general manner. . . . The varieties of physical phenomena that those sciences deal with have been found to correspond to discoverable varieties in the quantity of some variable. . . .

In these various transformations, the propositions of mathematics are merely doing what is proper to all propositions forming a train of reasoning—namely, enabling us to arrive indirectly, by marks of marks, at properties of objects that we can’t (or can’t so easily) ascertain by experiment. We travel from a given visible or tangible fact through mathematical truths to the facts that answer our questions. The given fact is a mark that a certain relation holds between the quantities Q_1 of some of the elements that are involved; the proposition that answers our question involves a certain relation between the quantities Q_2 of some other elements; if the quantities Q_2 are dependent in some known manner on Q_1 or vice versa, we can argue from the numerical relations between the quantities Q_1 to determine the relation that holds between the quantities Q_2, the links in the argument being provided by theorems of the calculus. And thus one physical fact becomes a mark of another by being a mark of a mark of a mark of it.

NOTES

[1] Because this is only an illustration, please allow me to assume that the basic principles of geometry are results of induction. I’ll try to prove this in chapter 5.

·MILL’S PROOF OF THE GEOMETRICAL THEOREM FROM HERE·

I’m working with an equilateral triangle whose vertices are A, D, E, with point B on the side AD, and point C on the side AE, such that BC is parallel to DE. We must begin, as in Euclid, by prolonging the equal sides AB and AC to equal distances, and joining the extremities BE and DC.

First formula: The sums of equals are equal.

AD and AE are sums of equals by the supposition. Having that mark of equality, they are concluded by this formula to be equal.

Second formula: When equal straight lines or angles are applied to one another, they coincide.

AC and AB are within this formula by supposition; AD, AE, have been brought within it by the preceding step. The angle at A considered as an angle of the triangle ABE, and the same angle considered as an angle of the triangle ACD, are of course within the formula. All these pairs, therefore, possess the property which, according to the second formula, is a mark that when applied to one another they will coincide. Conceive them, then, applied to one another, by turning over the triangle ABE, and laying it on the triangle ACD in such a manner that AB of the one shall lie upon AC of the other. Then, by the equality of the angles, AE will lie on AD. But AB and AC, AE and AD are equals; therefore they will coincide altogether, and of course at their extremities, D, E, and B, C.

Third formula: Straight lines, having their extremities coincident, coincide.

BE and CD have been brought within this formula by the preceding induction; they will, therefore, coincide.

Fourth formula: Angles, having their sides coincident, coincide.

The third induction having shown that BE and CD coincide, and the second that AB, AC, coincide, the angles ABE and ACD are thereby brought within the fourth formula, and accordingly coincide.

Fifth formula: Things which coincide are equal.

The angles ABE and ACD are brought within this formula by the induction immediately preceding. This train of reasoning being also applicable, mutatis mutandis [see Glossary], to the angles EBC, DCB, these also are brought within the fifth formula. And, finally,

Sixth formula: The differences of equals are equal.

The angle ABC being the difference of ABE, CBE, and the angle ACB being the difference of ACD, DCB; which have been proved to be equals; ABC and ACB are brought within the last formula by the whole of the previous process.

Chapter 5. Demonstration, and necessary truths

§1. If I have been right in chapters 3 and 4 in maintaining that

•Induction is the basis of all sciences, even the deductive or demonstrative ones,

•Every step in the ratiocinations even of geometry is an act of induction, and

•All that happens in a train of reasoning is that many inductions are brought to bear on a single subject of inquiry, using one induction to bring something within the range of another,

what are we to make of the sciences that are wholly or mostly deductive? What gives them the special certainty that is always ascribed to them? Why are they called the exact sciences? Why are the phrases ‘mathematical certainty’ and ‘demonstratively evident’ commonly used to express the highest degree of assurance that reason can attain? Why is it that almost all philosophers, and even some practitioners of the branches of natural science that have been converted into deductive sciences by the application of mathematics, •hold that mathematics is independent of the evidence of experience and observation and •regard it as a system of ‘necessary truth’?

The answer, I think, is that this character of necessity that is ascribed to the truths of mathematics is an illusion; and so is the special certainty that they are credited with (though I’ll later explain some reservations about this). In order to maintain this illusion, it is necessary to suppose that those truths concern the properties of purely imaginary objects. [Mill reminds us of his view that a definition can’t imply any proposition except one about the meaning of the defined expression, and that any further content that a ‘definition’ seems to have comes from its suppressed assumption that there are things answering to the expression as thus defined. He continues:] This assumption is not strictly true: there are no real things that exactly fit the definitions in geometry. There are no

•points with no size,

•lines with no breadth,

•perfectly straight lines,

•circles with all their radii exactly equal,

•squares with all their angles perfectly right-angled.

You may want to say that the assumption isn’t that such things actually exist but only that they could. I answer that, by any test we have of possibility, they couldn’t! So far as we can tell, their existence is inconsistent with the physical constitution of our planet, and perhaps the physical constitution of the universe. To remove this difficulty while holding onto the supposed system of ‘necessary truth’, it is usually said that the points, lines, circles, and squares that geometry deals with exist merely in our conceptions—they’re part of our minds. And our minds, by working on their own materials, construct an a priori science the evidentness of which •is purely mental and •has nothing whatever to do with outward experience. This doctrine may have been endorsed by some high authorities, but it appears to me psychologically incorrect. The points, lines, circles and squares that anyone has in his mind are (I submit) simply copies of the points, lines, circles and squares that he has known in his experience. Our idea of a point is simply our idea of the. . . .smallest portion of surface that we can see. A geometrically defined ‘line’ is wholly inconceivable. We can reason about a line as if it had no breadth; because when a perception is present to our senses, or a conception to our intellects, we can attend to a part of that perception or conception instead of to the whole. (Our ability to do that is the basis for all the control we can have over the operations of our minds.) But we can’t conceive of a line without breadth; we can’t form a mental picture of such a line; all the lines we have in our minds are lines with breadth. If you doubt this, think about your own experience! I really doubt that anyone who fancies that he can conceive a mathematical ‘line’ thinks so on the evidence of his consciousness; I suspect that it’s rather because he thinks that if such a conception were impossible mathematics couldn’t exist as a science. I’ll have no trouble showing that this is entirely groundless.

So the definitions of geometry don’t exactly correspond to anything in nature or in the human mind; but we can’t suppose that the subject-matter of geometry is nonentities. Our only option now is to consider geometry as dealing with lines, angles and figures that really exist; and its ‘definitions’ must be regarded as some of our first and most obvious generalisations concerning those natural objects. Those generalisations, considered just as generalisations, are flawlessly correct: the equality of all the radii of a circle is true of •all circles insofar as it is true of any •one; but it’s not exactly true of any circle; it is only nearly true—so nearly that we can pretend it to be exactly true without being led into any error of practical importance. When we have occasion to extend these inductions (or their consequences) to cases where the error would be appreciable—to lines of perceptible breadth or thickness, parallels which deviate perceptibly from equidistance, and the like—we correct our conclusions, by combining with them a fresh set of propositions relating to the aberration; just as we also take in propositions relating to the physical or chemical properties of the material, if those properties happen to make any difference to the result. . . . But as long as there’s no practical need to attend to any of the properties of the object except its geometrical properties, or to any of the natural irregularities in those, it is convenient to ignore the other properties and the irregularities in these ones, and to reason as if they didn’t exist; and we formally announce in the definitions that we intend to proceed on this plan. But our decision to confine our attention to just a few of an object’s properties doesn’t imply that we conceive or have an idea of the object denuded of its other properties! We are continually thinking of precisely the sorts of objects as we have seen and touched, and with all the properties they naturally have; but for scientific convenience we pretend they have been stripped of all properties except the ones that are relevant to our purpose.

So the unique accuracy the first principles of geometry are supposed to have appears to be fictitious. The assertions on which geometrical reasonings are based don’t exactly correspond with the facts, any more than do the bases of other sciences; but we suppose that they do, so as to trace the consequences of that supposition. Stewart’s is substantially right, I think, in his opinion about the foundations of geometry, namely

•that it is built on hypotheses;

•that this is the sole source of the special certainty it is supposed to have; and

•that in any science we can, by reasoning from a set of hypotheses, reach conclusions as certain as those of geometry;

i.e. as strictly in accordance with the hypotheses, and as irresistibly compelling assent on condition that the founding hypotheses are true.[1]

So when the conclusions of geometry are said to be ‘necessary truths’, their necessity consists only in their validly following from the suppositions from which they are deduced. ‘But aren’t those suppositions necessary?’ They’re not even true! They purposely depart, more or less widely, from the truth. The only sense in which the conclusions of any scientific investigation can be called ‘necessary’ is as a way of saying that they legitimately follow from some assumption which, by the conditions of that inquiry, is not to be questioned. . . . The conclusions of all deductive sciences were said by the ancients to be necessary propositions, but that’s because they didn’t understand the ‘not to be questioned’ status of the premises. . . .

§2. The important doctrine of Dugald Stewart that I have tried to secure has been contested by Whewell. . . . The supposed refutation of Stewart consists in proving against him (as I have also done here) that the premises of geometry are not •definitions but •assumptions of the real existence of things corresponding to those definitions. But those assumptions are the items that I call ‘hypotheses’, yet Whewell denies that geometry is founded on hypotheses. So he needs to show that the founding assumptions are absolute truths. But the furthest he goes in that direction is to say that at any rate they aren’t arbitrary hypotheses; that we aren’t at liberty to substitute other hypotheses for them; that. . . .the straight lines that we define, for instance, must be ‘those by which angles are contained, those by which triangles are bounded, those of which parallelism may be predicated, and the like’. This is true, but no-one has said that it isn’t. Those who say that the premises of geometry are hypotheses aren’t bound to maintain them to be hypotheses with no relation whatever to fact. An hypothesis framed for the purpose of scientific inquiry must relate to something x that has real existence (there can’t be a science of nonentities), so

•any hypothesis we make regarding x, to facilitate our study of it, mustn’t involve anything that is clearly false and in conflict with x’s real nature;

•we mustn’t ascribe to x any property that it doesn’t have;

•all we are free to do is to exaggerate slightly some of x’s properties (assuming it to be •completely what it really is •very nearly), and suppressing others.

•We’re absolutely obliged to restore the suppressed properties when their presence or absence would make a significant difference in the truth of our conclusions.

That’s the status of the first principles involved in the definitions of geometry. Is it necessary for a founding hypothesis to satisfy those constraints? Yes, if no other hypotheses could enable us to deduce conclusions which (with appropriate corrections) would be true of real objects; and the constraints can be brushed aside when our aim is only to •illustrate truths and not to •investigate them. We might suppose an imaginary animal and work out by deduction from the known laws of physiology its natural history; or an imaginary commonwealth, and from the elements composing it work out what would be its fate. Drawing conclusions from such purely arbitrary hypotheses might be a highly useful intellectual exercise; but the conclusions themselves could only teach us what would be the properties of objects that don’t really exist, so they don’t add anything to our knowledge of nature; whereas with an hypothesis that merely strips a real object of some of its properties, without clothing it in false ones, the conclusions will always express. . . .actual truth.

§3. But although Whewell hasn’t shaken Stewart’s doctrine about the hypothetical status of the first principles of geometry that are involved in the so-called ‘definitions’, I think he has greatly the advantage of Stewart on another important point—namely the necessity of including axioms as well as definitions among those first principles. Some of Euclid’s axioms could be exhibited in the form of definitions, or deduced by reasoning from propositions similar to ‘definitions’. Instead of the axiom

Magnitudes that can be made to coincide are equal,

we could introduce the definition

Equal magnitudes are those that may be so applied to one another as to coincide;

and then the three next axioms—

•Magnitudes that are equal to the same are equal to one another

•If equals are added to equals, the sums are equal

•If equals are taken from equals, the remainders are equal

—can be proved by an imaginary superposition like the one used in the demonstration of the fourth proposition of the first book of Euclid. But though these and several others may be deleted from the list of first principles because they can be demonstrated (though they don’t need to be), the list of axioms will still contain two or three fundamental truths that can’t be demonstrated. For example,

•Two straight lines can’t enclose a space

or its equivalent

•Straight lines that coincide at two points coincide altogether

and something to the effect that

•Two straight lines that intersect each other can’t both be parallel to a third straight line.[2]

Unlike the other class of fundamental principles that are involved in the definitions, the axioms—those that can’t be demonstrated as well as those that can—are true without any mixture of hypothesis. [He means: they are just true, not merely ‘true-if-hypothesis-H-is true’.] ‘Things that are equal to the same thing are equal to one another’ is just as true of the lines and figures in nature as it would be of the imaginary ones assumed in the definitions. It’s like this in most other sciences too. Almost all sciences have some general propositions that are •exactly true, while most are only •approximations to the truth. For example, •the first law of motion (the continuance of a movement until stopped or slackened by some resisting force) is true without qualification or error. •The rotation of the earth in twenty-four hours. . . .has gone on since the first accurate observations, without the increase or diminution of one second in all that period. These inductions don’t need any fiction—·any hypothesis·—to be accepted as accurately true; but there are others, such as the propositions about the shape of the earth, that are only approximations to the truth. To use them in the advance of our knowledge we have to pretend that they are exactly true, though they really fall short of that.

§4. What is the ground of our belief in axioms? What is our evidence for them? I answer: they are experimental truths, generalisations from observation. The proposition ‘Two straight lines can’t enclose a space’—or in other words ‘Two straight lines that have once met don’t meet again, but continue to diverge’—is an induction from the evidence of our senses.

This opinion goes against a scientific prejudice of long standing and great strength, and probably nothing that I say in this work will be as unfavourably received as this will. But it isn’t a new opinion; and even if it were, it should be judged not by its novelty but by the strength of the arguments in its favour. It is very fortunate that such an eminent defender of the contrary opinion as Whewell has presented an elaborate treatment of the whole theory of axioms, trying to construct the philosophy of the mathematical and physical sciences on the basis of the doctrine that I am now opposing. Anyone who wants to get to the bottom of a subject must rejoice to see the opposite side worthily represented. If what Whewell says in support of an opinion that he has made the foundation of a systematic work can be shown not to be conclusive, that will be enough—I shan’t need to look elsewhere for stronger arguments and a more powerful adversary!

I don’t have to show that the truths we call ‘axioms’ were first suggested by observation, and that we would never have known that two straight lines can’t enclose a space if we had never seen a straight line. Whewell admits this, as do all the recent supporters of his position. But they contend that experience doesn’t prove the axiom; that its truth is perceived a priori by the constitution of the mind itself, from the first moment when the meaning of the proposition is grasped, with no need to verify it by repeated trials as we have to do with truths that really are ascertained by observation.

But they can’t deny that the truth of the axiom ‘Two straight lines can’t enclose a space’, even if it is evident independently of experience, is also evident from experience. Whether or not the axiom •needs confirmation, it •receives confirmation in almost every instant of our lives: if we look at two straight lines that intersect one another we can’t help seeing that from the point of intersection they continue to diverge more and more. Empirical evidence crowds in on us in such endless profusion, with no cases where one might suspect an exception to the rule, that we would soon have stronger ground for believing the axiom than we have for almost any of the general truths that we confessedly learn from the evidence of our senses. [•Why ‘we would soon have. . . ’? Because Mill is saying ‘Even if experience were all we had to go on, we would. . . ’. He thinks that experience is all we have, but he is temporarily setting that aside and writing as though from Whewell’s point of view. •By ‘we confessedly learn. . . ’ he means that it’s generally agreed, not a matter of controversy, that we learn. . . .] Independently of a priori evidence, we would certainly believe it with an intensity of conviction far greater than we give to any ordinary physical truth; and we do this so early in our life that we don’t remember what was going on with us when we acquired this knowledge. So why assume that our recognition of these truths has a different origin from the rest of our knowledge, when its existence is perfectly accounted for by supposing its origin to be the same?. . . . The onus of proof lies on the advocates of the contrary opinion: it is for them to point out some fact that conflicts with the supposition that this part of our knowledge of nature has the same source as every other part.[3]

They could do this if they could prove that we had the conviction (at least practically) so early in infancy that we hadn’t yet received any of the sense-impressions that are the basis of the rival theory, ·the one I accept·. But no-one can prove this, because it concerns a time that is too far back for memory and too obscure for external observation. The friends of the a priori theory are obliged to rely on other arguments—basically just two of them. I’ll try to state them as clearly and forcibly as possible.

·FIRST ARGUMENT FOR A PRIORI GEOMETRICAL KNOWLEDGE·

§5. The first argument goes like this:

If our assent to the proposition that two straight lines can’t enclose a space were derived from the senses, the only way we could be convinced of its truth would be by actual trial, i.e. by seeing or feeling the straight lines; but in fact it is seen to be true by merely thinking of them. That a stone thrown into water goes to the bottom can be perceived by our senses, and we couldn’t be led to it by merely thinking of a stone thrown into water. It’s not like that with the axioms about straight lines: if I could be made to conceive what a straight line is, without having seen one, I would at once recognise that two such lines can’t enclose a space. Intuition is ‘imaginary looking’ (Whewell’s phrase) but experience must be real looking: if we see a property of straight lines to be true by merely imagining ourselves to be looking at them, the ground for our belief can’t be the senses or experience; it must be something mental.

Something that could be added to this argument for this particular axiom (not for all of them) is the following:

The evidence of eyesight isn’t merely unnecessary for this axiom; it is downright impossible. What does the axiom say? That two straight lines cannot enclose a space; that after intersecting once they will continue to diverge from one another even if they are prolonged to infinity. How could we, in any single case, see that this is so? However far we follow the lines, we can’t follow them to infinity; so we have to stop somewhere; and for all our senses can say to the contrary, the lines may just beyond our stopping-point begin to approach, and eventually to meet again. So if we didn’t have a proof of the impossibility other than what observation provides for us, we would have no ground for believing the axiom at all.

I don’t think I can be accused of understating these arguments.

A satisfactory answer to this line of thought will be found, I think, if we bring in one of the characteristic properties of geometrical forms—their ability to be depicted in the imagination with a distinctness equal to reality, i.e. the way our ideas of form exactly resemble the sensations that suggest them. This enables us (i) to make (at least with a little practice) mental pictures of all possible combinations of lines and angles, pictures that resemble the realities as well as any that we could make on paper; and (ii) to make those pictures just as fit subjects for geometrical experimentation as the realities themselves. ·How can that be? Well·, if the pictures are accurate enough, they will exhibit all the properties that the realities would show at one given instant and on simple inspection; and in geometry those are the only properties we are concerned with; we don’t care about the thing that pictures couldn’t exhibit, namely the inter-actions between bodies. So even if the ‘experiments’ (which in this case consist merely in attentive contemplation) were practised not on external objects but solely on what we call our ‘ideas’, i.e. on the diagrams in our minds, the foundations of geometry would still be laid in direct experience. In all systems of experimentation we take some objects to serve as representatives of everything that resembles them; and in our present case the conditions that qualify a real object to be the representative of its class are completely fulfilled by an object existing only in our imagination. Thus, without denying that we could satisfy ourselves that two straight lines can’t enclose a space by merely thinking of straight lines without actually looking at them, I contend that we don’t believe this truth simply on the basis of the imaginary intuition, but because we know that the imaginary lines exactly resemble real ones, and that we can conclude from

•imaginary lines to real ones

with as much certainty as we could conclude from

•one real line to another real line.

So the conclusion is still an induction from observation. And we wouldn’t be entitled to substitute observation of our mental image for observation of the reality if we hadn’t learned from long experience that the properties of the reality are faithfully represented in the image; just as we would be scientifically justified in describing an animal that we have never seen, on the basis of a picture made of it with a photograph; but only after we have learned from ample experience that observation of such a picture is precisely equivalent to observation of the original.

These considerations also remove the objection arising from the impossibility of looking the whole way along the lines as they extend to infinity. It’s true that we couldn’t actually see that two given lines never meet unless we followed them to infinity; but we do know that if they ever do meet—or if after diverging from one another they begin again to approach—this must happen at a finite distance from us. Supposing that it does happen, we can go there in imagination and form a mental image of the appearance that one or both of the lines must present at that point; and we can rely on that as being precisely similar to the reality. Now, whether we •focus on this imaginary picture or •remember the generalisations we have made from former views, we learn from our experience that a line which, after diverging from another straight line, begins to approach to it, produces the impression on our senses that we call ‘a bent line’, not ‘a straight line’.[4]

The preceding argument, which I think is unanswerable, merges in a still more comprehensive one that has been stated most clearly and conclusively by Bain. The psychological reason why axioms. . . .can be learned from the •idea only, without referring to the •fact, is that in the process of •acquiring the idea we have •learned the facts. . . . He writes:

‘We needed concrete experience in the first instance, to get the notion of whole and part; but once the notion is acquired it implies that the whole is greater than the part. In fact, we couldn’t have the notion without an experience tantamount to this conclusion. . . When we have mastered the notion of straightness, we have also mastered the aspect of it expressed by the statement that two straight lines can’t enclose a space. No intuitive or innate powers or perceptions are needed. . . We can’t have the full meaning of straightness without comparing straight objects with one another and with bent or crooked objects. One result of this comparison is that •straightness in two lines is seen to be incompatible with •enclosing a space; the enclosure of space involves crookedness in at least one of the lines.’

And similarly, in the case of every first principle, ‘the same knowledge that makes it understood, suffices to verify it’. The more this observation is considered the more (I am convinced) it will be felt to go to the very root of the controversy.

·SECOND ARGUMENT FOR A PRIORI GEOMETRICAL KNOWLEDGE·

§6. Now for the second argument in support of the theory that axioms are a priori truths. It goes like this:

Axioms are conceived by us not only as •true but as •universally and necessarily true. Now, experience can’t possibly tell us this about any proposition. I may have seen snow a hundred times, and seen that it was white, but this can’t give me entire assurance even that all snow is white, let alone that ·all· snow must be white.

[Mill continues with repetitions of this line of thought, quoted from Whewell, including:] ‘Experience. . . .contemplates external objects, but it can’t detect any internal bond that indissolubly connects the future with the past, the possible with the real. To learn a proposition by experience, and to see it to be necessarily true, are two altogether different processes of thought.’ And Whewell adds: ‘If anyone doesn’t clearly grasp this distinction between necessary and contingent truths, he won’t be able to join in our researches into the foundations of human knowledge—or indeed to pursue with success any speculation on the subject.’

What is the distinction the non-recognition of which incurs this denunciation? Whewell answers:

‘Necessary truths are those in which we not only learn that the proposition is true but see that it must be true; in which the negation of the truth is not only false but impossible; in which we can’t—even by an effort of imagination, or in a supposition—conceive the reverse of what is asserted. That there are such truths can’t be doubted: all relations of number, for example. Three and two added together make five. We can’t conceive it to be otherwise. We can’t by any freak of thought imagine three and two to make seven.’

Whewell has naturally and properly used a variety of phrases to bring his meaning more forcibly home, but I presume that he would allow that they are all equivalent, and that what he means by ‘a necessary truth’ would be sufficiently defined as ‘a proposition the negation of which is not only false but inconceivable’. I can’t find in any of his expressions. . . .a meaning beyond this, and I don’t think he would contend that they mean anything more.

So this is the principle asserted: that if the negation of proposition P is inconceivable,. . . .P must rest on evidence of a higher and more forceful description than any that experience can provide.

I’m surprised that so much stress should be laid on inconceivability, when there’s so much empirical evidence that our (in)ability to conceive x has very little to do with x’s possibility, and a great deal to do with. . . .the past history and habits of our own minds. We find it extremely difficult to conceive as possible something that contradicts long established and familiar experience, or even merely old familiar habits of thought. Everyone knows this; it’s a necessary result of the fundamental laws of the human mind, specifically of the primary law of association. When we have often seen and thought of two things together, and have never seen or thought of them separately, there’s an increasing difficulty—which may in the end become insuperable—of conceiving them apart. This is most conspicuous in uneducated persons, who are in general utterly unable to separate any two ideas that have become firmly associated in their minds; and if persons with developed intellects do any better in this, it’s only because—having seen and heard and read more, and being more accustomed to exercise their imagination—they have experienced their sensations and thoughts in more varied combinations, which has prevented them from forming many of these inseparable associations. But. . . .·even· the most practised intellect is not exempt from the universal laws of our conceptive faculty. If daily habit presents to someone for a long time two facts in combination, and if he isn’t led during that period. . . .to •think of them apart, he will probably in time become unable to •do that even by the strongest effort; and the supposition that the two facts can be separated in nature will eventually present itself to his mind with all the marks of an inconceivable phenomenon. There are remarkable examples of this in the history of science: cases where highly educated men rejected as impossible, because inconceivable, things that their posterity. . . .found it quite easy to conceive and that everybody now knows to be true. There was a time when men with the most cultivated intellects and the greatest freedom from the domination of early prejudice couldn’t believe in the existence of antipodes, because they couldn’t conceive the force of gravity acting upward instead of downward. The Cartesians long rejected the Newtonian doctrine of the gravitation of all bodies toward one another on the strength of the proposition that a body can’t •act where it •is not, the reverse of which seemed to them inconceivable. All the cumbrous machinery of imaginary vortices [see Glossary], assumed with no evidence, appeared to these philosophers a more rational account of the heavenly motions than one involving what seemed to them such a great absurdity.[5]

No doubt they found it as impossible to conceive (a) a body acting on the earth from the distance of the sun or moon as we find it to conceive (b) an end to space or time, or (c) two straight lines enclosing a space. Newton himself wasn’t able to conceive (a), which is why we have his hypothesis of a ‘subtle ether’, the hidden cause of gravitation; and his writings show that although he regarded the particular nature of the intermediate agency to be a matter of conjecture, he had no doubt that there must be some such agency.

If it’s so natural to the human mind, even in a high state of culture, to •be unable to conceive and therefore to •believe impossible something that is later found to be not only conceivable but true, it’s not surprising that in cases where

the association is still older, more confirmed, and more familiar, and nothing ever occurs to shake our conviction, or even suggest to us any conception at variance with the association,

the •acquired incapacity continues and is mistaken for a •natural incapacity! Our experience of the varieties in nature does enable us, within limits, to conceive other varieties analogous to them. . . . But when experience affords no model on which to shape the new conception, how could we form it? How can we imagine an end to space or time? We never saw an object without something beyond it, or experienced a feeling without something following it. So when we try to conceive the last point of space, the idea of other points beyond it arises irresistibly. When we try to imagine the last instant of time, we can’t help conceiving another instant after it. There is no need to assume, as do the Kantians, a peculiar fundamental law of the mind to account for the feeling of infinity inherent in our conceptions of space and time; that apparent infinity is well enough accounted for by simpler laws that everyone accepts.

Now turn back to geometrical axioms such as Two straight lines can’t enclose a space—a truth that is confirmed by our very earliest impressions of the external world—how could its falsity possibly be conceivable to us?. . . . What analogy do we have, what similar order of facts in any other branch of our experience, to help us to conceive two straight lines enclosing a space? Also, remember my point that our ideas or mental images of form exactly resemble the things they are ideas of, and represent them well enough for the purposes of scientific observation. From this, and from the intuitive character of the observation (which in this case reduces itself to simple inspection), we can’t call up in our imagination two straight lines, so as to try to conceive them enclosing a space, without by that very act repeating the scientific experiment that establishes the contrary. Will it really be contended that in a case like this the inconceivability of the axiom’s falsity is evidence against the thesis that our belief in the axiom has an empirical origin? Isn’t it clear that however that belief originated, the impossibility of our conceiving the negative of it will be the same? Whewell urges those who have difficulty recognising his distinction between necessary and contingent truths to study geometry, and I can assure him that I have conscientiously done that. Now I in return, with equal confidence, urge those who agree with him to study the general laws of association. I’m convinced that a moderate familiarity with those laws is all that is needed to dispel the illusion that ascribes a special necessity to our earliest inductions from experience, and measures the •possibility of things in themselves by •the human ability to conceive them.

Whewell himself has both (a) confirmed by his testimony the effect of habitual association in making an empirical truth appear to be necessary, and (b) provided a striking instance of the law of in his own person.

·(a) WHEWELL IMPLYING THAT ASSOCIATIONS CREATE INCONCEIVABILITY·

In his Philosophy of the Inductive Sciences Whewell repeatedly says something striking about now-established propositions that we know were discovered gradually and by great efforts of genius and patience. Once they are established, he says, we find it •hard to conceive that they weren’t recognised from the outset by everyone whose mind was in good order. If we didn’t know the history of their discovery we would find it •impossible.

‘We now despise the opponents of Copernicus who couldn’t conceive the sun’s appearing to move when really it doesn’t;. . . .the opponents of Newton who held there was something absurd in his doctrine that differently coloured light-rays are refracted at different angles; those who thought that when elements combine, their sensible qualities must show up in the compound; and those who were reluctant to give up the classification vegetables into herbs, shrubs, and trees. We can’t help thinking that men must have been thick-headed to find a difficulty in admitting what is to us so plain and simple. . . . But most of those people who were on the losing side were no more prejudiced or stupid or narrow-minded than most of us are today; and the side they backed was far from being obviously wrong until it had been condemned as wrong by the result of the war ·between theories·. . . In most of these cases the victory of truth has been so complete that now we can hardly imagine the struggle to have been necessary. The very essence of these triumphs is that they lead us to regard the views we reject as not only false but inconceivable.’

This last proposition is precisely what I contend for; and it’s all I need to overthrow Whewell’s theory about the nature of the evidentness of axioms. For what is that theory? That the truth of axioms can’t have been learned from experience because their falsity is inconceivable. But Whewell himself says that we’re continually led in the natural progress of thought to regard as inconceivable propositions that our forefathers not only conceived but believed—and indeed (he might have added) were unable to conceive to be false. . . . After such a complete admission that inconceivability is an accidental thing, not inherent in the phenomenon itself but dependent on the mental history of the person who tries to conceive it, how can he ever call on us to reject a proposition as impossible simply because it is inconceivable? Yet he does so; and along the way he unintentionally provides some very remarkable examples of the very illusion—·the illusion of inherent inconceivability·—that he has himself so clearly pointed out. I select his remarks on the evidentness of the three laws of motion, and of the atomic theory.

With respect to the laws of motion, Whewell says:

‘No-one can doubt that in historical fact these laws were collected from experience. This isn’t a mere conjecture. We know the time, the persons, the circumstances, belonging to each step of each discovery.’

. . . .And not only were these laws far from intuitively evident, but some of them were originally paradoxes. The first law was especially so. A moving body will continue moving at the same speed for ever unless some new force acts on it—this was a proposition that mankind for a long time found almost impossible to believe. It went against apparent experience of the most familiar kind, which taught that it was the nature of motion •to lessen gradually and at last •to stop. Yet once the contrary doctrine was firmly established, Whewell points out, mathematicians rapidly began to believe that these laws—contradictory to first appearances, and hard to make familiar to the minds of the scientific world even after full proof had been obtained—were under ‘a demonstrable necessity, compelling them to be such as they are and no other’; and Whewell himself, though not venturing ‘absolutely to pronounce’ that all these laws ‘can be rigorously traced to an absolute necessity in the nature of things’, does have that view of the law I have mentioned, of which he says:

‘Though the discovery of the first law of motion was made, historically speaking, by means of experiment, we have now attained a point of view in which we see that it could have been certainly known to be true, independently of experience.’

Can there be a more striking instance than this of the effect of association that I have described? Philosophers for generations have tremendous difficulty putting certain ideas together; they at last succeed in doing so; and after repeating this process often enough they first imagine a natural bond between the ideas, and then experience a growing difficulty, which eventually grows to an impossibility, of separating them from one another. If that’s what happens to empirical beliefs that began only yesterday and are in opposition to first appearances, how must it fare with beliefs that square with appearances that are familiar from the first dawn of intelligence. . . .?

·(b) WHEWELL ILLUSTRATING IN HIMSELF THE POWER OF ASSOCIATIONS TO CREATE INCONCEIVABILITY·

In discussing the atomic theory Whewell provides a truly astonishing example—it could be called the reductio ad absurdum [see Glossary] of the theory of inconceivability. Speaking of the laws of chemical composition, he says: ‘It’s certain that these laws could never have been clearly understood, and therefore never firmly established, without laborious and exact experiments. But I venture to say that once they are known, they have an evidentness that mere experiment could never provide. For how in fact can we conceive of combinations otherwise than as definite in kind and quality? If we were to suppose each element ready to combine with any other indifferently, and indifferently in any quantity, we would have a world where all would be confusion and indefiniteness. There would be no fixed kinds of bodies. Salts, and stones, and ores would gradually shade into each other. But we know that the world consists of bodies distinguishable from each other by definite differences, capable of being classified and named, and of having general propositions asserted about them. And as we can’t conceive a world that is not like that, it seems that we can’t conceive a state of things in which the laws of the combination of elements should not be of that definite and measured kind that I have been discussing. That a philosopher of Whewell’s eminence should gravely assert that we can’t conceive a world in which the simple elements combined in other than definite proportions; that by meditating on a scientific truth, the original discoverer of which was still living, he made the association in his own mind between the idea of •combination and the idea of •constant proportions so familiar and intimate that he can’t conceive of one fact without the other; is such a striking instance of the mental law that I am defending that there’s no need for me to offer a word of comment on it!

[Mill now reports a move that Whewell makes in his most recent writings on this topic. He says that the necessity of the atomic theory is merely something that ‘philosophical chemists in a future generation may possibly see’. And that what he is talking about is the inability to conceive something ‘distinctly’: something that is really impossible may be (vaguely) conceived by the man in the street; and it may be (a little vaguely) conceived by a scientist until he gets up to the level of finding it inconceivable. Thus:] Necessary truths are not those of which we can’t conceive the contrary, but those of which we can’t distinctly conceive the contrary. . . . By the ever-increasing distinctness with which scientific men grasp the general conceptions of science, they eventually come to perceive that there are certain laws of nature which, though as a matter of fact they were learned from experience, we can’t, now that we know them, distinctly conceive to be other than they are. [This paragraph has been entirely a report on Whewell.]

I would give a somewhat different account of this progress of the scientific mind. After a general law of nature has been ascertained, men’s minds don’t right away become easily able to think of natural phenomena strictly in terms of it. The habit that constitutes the scientific cast of mind—

the habit of conceiving facts of all kinds in ways that square with the laws that regulate them, conceiving phenomena of all kinds according to the relations that have been found really to exist between them

—this habit, in the case of newly-discovered relations, comes gradually; and until it is thoroughly formed, the new truth isn’t regarded as necessary. But eventually the philosopher [here = ‘thoughtful scientist’] achieves a state of mind in which his mental picture of nature spontaneously represents to him all the phenomena that the new theory deals with in exactly the light in which the theory regards them; all images or conceptions derived from any other theory, or from the confused view he had before he had any theory, entirely disappear from his mind. The way of representing facts that results from the theory has now become the only way of conceiving them that he finds natural. A prolonged habit of arranging phenomena in certain groups, and explaining them by means of certain principles, makes any other arrangement or explanation of these facts feel unnatural: and he may eventually find it as difficult to represent the facts to himself in •any other way as it used to be to represent them in •that way.

. . . .A contradiction is always inconceivable; so our scientist’s imagination rejects false theories and says it can’t conceive them. But their inconceivability to him doesn’t result from anything in the theories themselves, any inherent conflict with the human mind; it results from the conflict between them and some of the facts; and the scientist found the false theory conceivable as long as he didn’t know those facts. . . . So his real reason for rejecting theories at variance with the true one is just that they clash with his experience, but he easily slides into believing that he •rejects them because they are inconceivable, •adopts the true theory because it is self-evident, and •has no need for it to be made evident by experience.

I think this is the real explanation of the paradoxical truth stressed by Whewell, that having a scientifically cultivated mind makes one unable to conceive suppositions that a common man conceives with no difficulty. There’s nothing inconceivable in the suppositions themselves. . . . In the case of many of Whewell’s ‘necessary truths’ the negative of the axiom is as easily conceivable as the affirmative, and will probably be so as long as the human race lasts. Consider the axiom that matter is indestructible, which is as high as anything on Whewell’s list of propositions that are necessary and self-evident. I quite agree that this is a true law of nature, but I don’t think that anyone has any difficulty in ·conceiving or· imagining a portion of matter being annihilated. It wouldn’t have to look different from events we see all the time—water drying up, fuel being consumed. And the law that bodies combine chemically in definite proportions is undeniably true; but few people have reached the point that Whewell seems personally to have arrived at, of being unable to conceive a world in which the elements combine with one another ‘indifferently in any quantity’. Whewell dares to prophesy similar success to the multitude only after the lapse of generations; but it’s not likely that we’ll ever rise to this sublime height of inability, so long as all the mechanical mixtures in our planet—whether solid, liquid, or gaseous—exhibit to our daily observation the very phenomenon declared to be inconceivable.

Whewell says that these and similar laws of nature can’t be drawn from experience because they are assumed in the interpretation of experience. Our inability to ‘add to or diminish the quantity of matter in the world’ is a truth that ‘neither is nor can be derived from experience; for the experiments we make to verify it presuppose its truth. . . When men began to use the balance in chemical analysis, they. . . .but took for granted as self-evident that the weight of the whole must be the sum of the weights of the elements.’ True, it is assumed; but only in the way that all experimental inquiry provisionally assumes some theory or hypothesis, which is to be finally accepted or not according as the experiments decide. The hypothesis. . . .that the material of the world, as estimated by weight, is neither increased nor diminished by any of the processes of nature or art, had many appearances in its favour to begin with. . . . There were other facts that appeared to conflict with it, so experiments were devised to verify it. Men assumed its truth hypothetically, and proceeded to try whether the phenomena that apparently pointed to a different conclusion would on further investigation be found to be consistent with it. This turned out to be the case; and from then on the doctrine took its place as a universal truth—proved to be such by experience. That the theory itself preceded the proof of its truth—that it had to be conceived before it could be proved—doesn’t imply that it was self-evident and didn’t need proof. Otherwise all the true theories in the sciences are necessary and self-evident; for no-one knows better than Whewell that they all began by being assumed, for the purpose of connecting them by deductions with the facts of experience that now count as evidence in their favour.

·LONG FOOTNOTE APPENDED TO CHAPTER 5·

The Quarterly Review for June 1841 contained a very able article on Whewell’s two great works, an article that maintains on the subject of axioms the doctrine I have been defending here—that axioms are generalisations from experience—and supports that opinion by a line of argument strikingly like mine. Nearly all the present chapter was written before I had seen the article, but in saying this I’m not claiming originality but merely calling your attention to the fact that two inquirers have, entirely independently of one another, arrived at an opinion that is opposed to reigning doctrines. I’m glad to have this opportunity to quote passages that are remarkably in unison with my own views—passages written by someone whose extensive physical and metaphysical knowledge, and capacity for systematic thought, are shown by the article. [The writer was Sir John Herschel.]

‘The truths of geometry are summed up and embodied in its definitions and axioms. . . Let us turn to the axioms, and what do we find? A string of propositions concerning magnitude in the abstract, which are equally true of space, time, force, number, and every other magnitude that can be added and divided. Such propositions, apart from those that are not mere definitions, carry their inductive origin on the surfaces of the sentences expressing them. . . The only ones that express characteristic properties of space are “Two straight lines can’t enclose a space” and “Two straight lines that intersect can’t both be parallel to a third”. Let us have a closer look at these. The only clear notion we can form of •straightness is •uniformity of direction, for space in the final analysis is nothing but an assemblage of distances and directions. And. . . .we can’t even make the proposition intelligible to anyone whose experience ever since he was born hasn’t assured him of the fact. The unity of direction—i.e. that we can’t march straight from x to y by more than one route is matter of practical experience long before it could possibly be matter of abstract thought. We can’t attempt to imagine a situation in which it would be false, without •violating our habitual recollection of this experience and •defacing the mental picture of space that we have based on it. What other than experience could possibly assure us of the homogeneity of the parts of distance, time, force, and measurable aggregates in general, on which the truth of the other axioms depends?. . . .’

Concerning axioms of mechanics: ‘. . . .Let us take one of these axioms and ask what makes it evidently true: for instance, that equal forces perpendicularly applied at the opposite ends of equal arms of a straight lever will balance each other. What other than experience can possibly inform us that a force so applied will have any tendency to turn the lever on its centre at all? or that force can be transmitted along a rigid line perpendicular to its direction in such a way as to act at a place that isn’t along its own line of action? This is so far from being •self-evident that it even seems •paradoxical until we bring in the lever’s thickness, material composition, and molecular powers. Again, we conclude that the two forces, being equal and applied under precisely similar circumstances, must if they exert any effort at all to turn the lever exert equal and opposite efforts; but what a priori reasoning can possibly assure us that they do act under precisely similar circumstances? that their being in different places doesn’t affect the forces that they exert? [The argument continues, in the spirit of Mill’s discussion. Then a further axiom:] The other fundamental axiom of statics, that the pressure on the point of support is the sum of the weights. . . is merely a scientifically more refined statement of a coarse and obvious result of universal experience, namely that the weight of a rigid body is the same, however we handle it or suspend it, and that whatever holds it up holds up its total weight. Whewell rightly says: ‘Probably no-one ever did an experiment to show that the pressure on the support is equal to the sum of the weights.’. . . But that’s because in every action of someone’s life from earliest infancy he is continually doing the experiment and seeing it done by every other living being about him, so that he never dreams of staking its result on one additional attempt made with scientific accuracy. This would be like sealing yourself up for half an hour in a metal case so as to discover whether your eyes are useful for seeing.’

On the ‘paradox of universal propositions obtained by experience’ the writer says: ‘If there are necessary and universal truths expressible in propositions of axiomatic simplicity and obviousness, and having for their subject-matter the elements of all our experience and all our knowledge, surely these are the truths that experience ought to suggest most readily, clearly, and unceasingly. If it were a universal and necessary truth that a net is spread over the whole surface of every planetary globe, we wouldn’t travel far on our own without getting entangled in its meshes, and making the necessity of some means of extrication [i.e. our need for some such means] an axiom of locomotion! So there is nothing paradoxical—quite the reverse—in our being led by our senses to a recognition of such truths as general propositions that are at least true of all human experience. That •they pervade all the objects of experience ensures their continually being suggested by experience; that •they are true ensures the consistency of suggestion. . . that commands complete assent; that •they are simple and can’t be misunderstood secures their admission by every mind.’

‘A necessary and universal truth about any object of our knowledge must verify itself in every state of affairs where we are thinking about that object, and if at the same time it is simple and intelligible, its verification must be obvious. Thus the sentiment of such a truth can’t not be present to our minds whenever that object is contemplated, and must therefore be part of the mental picture or idea of that object that we may sometimes bring before our imagination. . . That’s why all propositions become not only untrue but inconceivable if. . . axioms are violated in the statement of them. . . .’

·END OF LONG FOOTNOTE·

NOTES

[1] Bain rightly says that ‘hypothesis’ is being used here in a somewhat peculiar sense. When in science something is called an ‘hypothesis’ this usually means that it is not known to be true but is surmised to be so because that would account for certain known facts; and the final result of the theoretical inquiry may be to prove its truth. The hypotheses I have spoken of here are not like that; they are known not to be literally true, and as much of them as is true is not hypothetical but certain. With ‘hypotheses’ in either sense, however, we reason not from a truth but from an assumption, and the truth therefore of the conclusions is conditional, not categorical. That is enough to justify. . . .Stewart’s use of the term. But we mustn’t forget that the hypothetical element in a geometrical definition is the assumption that what is •very nearly true is •exactly so. This unreal exactitude might be called ‘a fiction’ as properly as ‘an hypothesis’; but ‘fiction’ would be even further than ‘hypothesis’ would from reminding us of how closely the fictitious point or line is related to the points and lines of which we have experience.

[2] We could insert this last property into the definition of ‘parallel’, making it require both that (i) when produced indefinitely the lines will never meet and (ii) that any straight line that intersects one of them will, if prolonged, meet the other. But this still doesn’t remove the need for the assumption; we are still obliged to take for granted the geometrical truth that all straight lines in the same plane that have property (i) also have (ii). For if that weren’t so, the demonstrations of later parts of the theory of parallels would fail.

[3] Some writers say that ‘Two straight lines can’t enclose a space’ couldn’t become known to us through experience for this reason:

If the straight lines in question are absolutely without breadth and absolutely straight, experience can’t show us that they can’t enclose a space because we don’t have any experience of such lines. And if the lines are of the sort we do meet with in experience—straight enough for practical purposes but actually slightly zigzag, and with some thickness—the axiom isn’t true, for two of those lines can and sometimes do enclose a small portion of space. In neither case, therefore, does experience prove the axiom.

Those who use this argument to show that geometrical axioms can’t be proved by induction show themselves unfamiliar with a common and perfectly valid mode of inductive proof—namely proof by approximation. Though experience doesn’t present lines so unimpeachably straight that two of them are incapable of enclosing the smallest space, it presents us with gradations of lines having less and less thickness or zigzagging; this constitutes a series of which the ‘straight line’ of the geometrical definition is the ideal limit. What observation shows us is that the nearer the straight lines of experience come to having no thickness and no kinks, the nearer the space-enclosing power of any two of them comes to zero. The inference that if they had no thickness or kinks at all they would enclose no space at all is a correct inductive inference from these facts. It fits one of the four inductive methods that I’ll present in III.8, namely the Method of Concomitant Variations. . . .

[4] Whewell thinks it unreasonable to contend that we know by experience that our idea of a line exactly resembles a real line. He writes: ‘I don’t see how we can compare our ideas with the realities, given that we know the realities only by our ideas.’ We know the realities by our sensations. Whewell surely doesn’t hold the ‘doctrine of perception by means of ideas’ which Reid took so much trouble to refute. If Whewell doubts whether we compare our ideas with the corresponding sensations and assume that they are alike, let me ask him: Why do we judge that a portrait of someone not present is like the original? Surely because it is like our idea or mental image of the person, and because that idea is like the man himself.

Whewell also challenges the thesis that this resemblance of ideas to the sensations of which they are copies is a special feature of ideas of space. I reply that I assert no such thesis. Ideas of space are special only in how closely and exactly they resemble the corresponding sensations. No-one would claim to imagine a colour or odour as closely and accurately as almost everyone can mentally reproduce an image of a straight line or a triangle. [Mill goes on to say that the imagining of colours or odours can be put to use. ‘Which has the darker blue—the flower that I gave you a week ago or the one I put on your father’s grave last month?’ Someone might be able to answer this by comparing his mental pictures of the flowers and reading off his answer from that comparison. He might, but, Mill continues:] People differ widely in how precisely they can recollect things: one person, when he has looked someone in the face for half a minute, can draw an accurate likeness of him from memory; another may have seen him every day for six months yet hardly know whether his nose is long or short. But everyone has a perfectly distinct mental image of a straight line, a circle, or a rectangle. And everyone confidently argues from these mental images to the corresponding outward things. We can and continually do study nature in our recollections, when the objects themselves are absent; and in the case of geometrical forms we can perfectly trust our recollections, while in most other cases we can trust them only imperfectly.

[5] It would be hard to name a man more remarkable for the •greatness and the wide •range of his mental accomplishments than Leibniz. Yet this eminent man rejected Newton’s account of the solar system on the grounds that God could not make a body revolve round a distant centre except by some mechanism that pushed them into moving like that, or by miracle. He wrote to the Abbé Conti: ‘Anything that can’t be explained by the nature of created things is miraculous. It won’t do to say “God has made such-and-such a law of nature, so the thing is natural.” The law has to be something that can be carried out by the natures of created things. For example, if God gave a law compelling a free body to turn around a certain centre, he would have to •connect it with other bodies which by pushing it forced it always to stay in its circular orbit, or •give that job to an angel, or •do it by a miracle; because what the body would do naturally is to leave the orbit along a tangent.’

Chapter 6. Demonstration and necessary truths (cont’d)

§1. The discussion in chapter 5 of the deductive sciences that are commonly said to be systems of necessary truth has led to the following conclusions. The results of those sciences are indeed ‘necessary’ in the sense of ‘necessarily following from “axioms” and definitions’, i.e. of being certainly true if those axioms and definitions are true. (Even in this sense, you see, ‘necessity’ simply means ‘certainty’.) But if any scientific result R is to count as necessary in any sense beyond this—any sense implying that R is evidently true in a way that doesn’t depend on observation and experience—we must first establish that the definitions and axioms from which R is inferred are themselves necessary in that sense. ·And we have seen that this can’t be established because it isn’t true·. I have shown that axioms considered as experimental [= ‘empirical’?] truths rest on superabundant and obvious ·empirical· evidence. I then asked:

Are we then compelled to suppose •that such ‘axioms’ are evident because of something other than experimental evidence, •that our acceptance of them has a non-empirical basis?

I argued that if anyone answers Yes, the burden of proof lies with him; and I thoroughly examined the arguments they have produced. These all failed the test, which I took as a justification for concluding that axioms are merely one class, the most universal class, of inductions from experience—the simplest and easiest cases of generalisation from the facts delivered by our senses or by our internal consciousness.

In contrast with that, the improperly so-called ‘definitions’ in demonstrative sciences turned out to be generalisations from experience that aren’t even truths, strictly speaking. Why? Because

They are propositions in which we •assert of some kind of object that it has some property or properties that observation shows to belong to it, but also •deny that it has any other properties, though in each individual instance the thing does have other properties, nearly always ones that modify [see Glossary] the one that is asserted in the definition.

The point is that the denial is a mere fiction—a supposition— made for the purpose of •excluding the consideration of those details when their influence is of too trifling amount to be worth considering, or (if it’s not trivial) postponing it to a more convenient moment. [The word ‘details’ replaces Mill’s ‘modifying circumstances’—see ‘modify’ in the indented passage.]

From all this it seems that deductive or demonstrative sciences are all inductive sciences; that what makes them evident is the evidentness of experience; but they are also hypothetical sciences because of the special character of one indispensable ingredient in the general formulae according to which their inductions are made. Their conclusions are true only on certain suppositions, which are or ought to be approximations to the truth, but are seldom if ever exactly true. And this hypothetical character is the real source of the special certainty that is supposed ·by some theorists· to be inherent in demonstration!

The position I have been defending, however, has no chance of being accepted as true of all deductive or demonstrative sciences until it has been checked against the most remarkable of all those sciences, that of numbers—i.e. the theory of •the calculus, •arithmetic and •algebra. It’s harder to believe of the doctrines of this science than of any other •that they are not known a priori but are experimental truths, or •that their special certainty comes from their not being absolute but only conditional truths. [Just to make sure that the last clause is understood: however great the chance is that a proposition Q (absolute) is true, there may be a much better chance (and there can’t be a worse one) that ‘If P then Q’ (conditional) is true.] So the science of numbers needs to be examined separately, especially given that on this subject two opposing doctrines have to be dealt with: •that of the a priori philosophers and •a second one that is the most opposite to theirs, used to be very generally accepted, and is still far from being altogether exploded among metaphysicians.

§2. This ·second· theory ·is called ‘nominalism’ (from the Latin nomen = ‘name’); it· represents the propositions of the science of numbers as merely verbal, and its processes as mere substitutions of one expression for another. The proposition ‘Two and one is equal to three’, according to these writers, is not a truth, is not the assertion of a really existing fact, but a definition of ‘three’; a statement that mankind have agreed to use ‘three’ as a short name for anything that is called by the clumsy phrase ‘two and one’. According to this doctrine, even the longest calculation in algebra is just a series of changes in terminology, in which equivalent expressions are substituted one for another—a series of translations of a single fact from one language into another—though the friends of this theory haven’t explained how such a series of translations can have as output a different fact, as when we demonstrate a new geometrical theorem by algebra; and this failure is fatal to their theory.

It must be acknowledged that the processes of arithmetic and algebra have some special features that make this theory in question very plausible, and have naturally made those sciences the stronghold of nominalism. The doctrine that

we can discover facts—detect the hidden processes of nature—by a skillful manipulation of language

is so contrary to common sense that a person must have made some advances in philosophy to believe it! What drives people to this paradoxical belief is their perceived need to avoid some even greater difficulty that the vulgar [see Glossary] don’t see. Many people have come to think that reasoning is a merely verbal process because no other theory seemed reconcilable with the nature of the science of numbers. The facts about that science that have impressed them are these:

When we use the symbols of arithmetic or of algebra, we don’t carry any ideas along with us. In a geometrical demonstration we do have a diagram, in our minds if not on paper, so that (for example) AB, AC are present to our imagination as lines that intersect and form an angle. But not so with the a and b of algebra: they can represent lines or any other magnitudes, but those magnitudes are never thought of; nothing on show in our imagination but a and b. The ideas that they happen to represent on the particular occasion are banished from the mind throughout the process between •the start where the premises are translated from things into signs and •the end where the conclusion is translated back from signs into things.

Given that there is nothing in the reasoner’s mind but the symbols, how can the reasoning process be concerned with anything other than the symbols?. . . .

But when we think about it we’ll see that •this apparently decisive instance is really not an instance at all; that •in every step of an arithmetical or algebraic calculation there’s a real induction, a real inference of facts from facts; and that •what disguises the induction is simply its comprehensive nature and thus the extreme generality of the language it uses. All numbers must be numbers of something: there are no such things as numbers in the abstract. Ten must mean ten bodies, or ten sounds, or ten beats of the pulse. But they can be numbers of anything. So propositions about numbers have the remarkable special feature that they are propositions about everything—all objects, all existents of every kind that we have encountered. All things have quantity, consist of parts that can be numbered, so they have all the properties that are called properties of numbers. That half of four is two must be true whatever ‘four’ represents, whether four hours, four miles, or four pounds weight. We need only to conceive a thing as divided into four equal parts (and everything can be conceived as so divided) to be able to predicate of it every property of the number four, i.e. every arithmetical proposition in which the number four stands on one side of the equation. Algebra extends the generalisation still further: every number represents that particular number of all things without distinction, but every algebraic symbol represents all numbers without distinction. As soon as we conceive a thing divided into equal parts, without knowing how many parts, we can call it a or x, and apply to it, with no risk of error, every algebraic formula in the books. The proposition 2(a + b) = 2a + 2b is a truth throughout all nature. And because algebraic truths are true of all things whatever. . . ., it’s no wonder that the symbols don’t arouse in our minds ideas of anything in particular. . . . We don’t need under the symbol a to picture to ourselves all things whatever, but only some one thing, ·any one thing·; so why not the letter itself? The mere written characters, a, b, x, y, z represent things in general as well as any more complex and apparently more concrete conception. What shows that we are conscious of them as •things and not as mere •signs is the fact that throughout our reasoning we predicate of them the properties of •things. In solving an algebraic equation, by what rules do we proceed? By applying at each step to a, b, and x the propositions

•that equals added to equals make equals;and

•that equals taken from equals leave equals;

and other propositions based on those two. These aren’t properties of language, or of signs as such, but of magnitudes, which amounts to saying of all things. So the successive inferences concern things, not symbols. And although any things whatever will serve our purpose, there’s no need for us to keep the idea of the thing at all distinct; and because of that, there’s no risk in allowing our process of thought to become entirely mechanical, which is what thought-processes do (if permitted) when they are performed often.

Thus the general language of algebra comes to be used familiarly without arousing ideas. All general language is apt to do this from mere habit, though algebra is the only context where it can be done with complete safety. But when we look back to see what gave the process its force as a proof, we find that every single step brings us along only if we suppose ourselves to be thinking and talking about the things and not the mere symbols.

The notion that the propositions of arithmetic and algebra are merely verbal gets even more plausibility from something else. It’s the fact that when propositions are considered as being about things, they all look like identical propositions [see Glossary]. Consider ‘Two and one is equal to three’, considered as an assertion about objects—e.g. ‘Two pebbles and one pebble are equal to three pebbles’. This doesn’t affirm •equality between two collections of pebbles but •absolute identity. It affirms that if we add one pebble to two pebbles those very pebbles are three. We have the same objects throughout, and the mere assertion that ‘objects are themselves’ is empty, so it seems only natural to regard ‘Two and one is equal to three’ as merely asserting that the two names have the same signification.

Plausible as this looks, it won’t bear examination. The expression ‘two pebbles and one pebble’ and the expression ‘three pebbles’ do indeed stand for the same collection of objects, but they don’t stand for the same physical fact. They’re names of the same objects, but of those objects in two different states: though they •denote the same things, their •meaning is different. Three pebbles in two separate parcels don’t make the same impression on our senses as three pebbles in one parcel; and the assertion that the very same pebbles can be arranged so as to produce either of those sets of sensations, though a very familiar proposition, is not an identical one. It is a truth known to us by early and constant experience, an inductive truth; and such truths are the foundation of the science of number. The fundamental truths of that science all rest on the evidence of the senses; they are proved by showing to our eyes and fingers that any given number of objects—ten balls, say—can by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children are based on a knowledge of this fact. All who wish to carry the child’s mind along with them in learning arithmetic—all who wish to teach numbers and not mere ciphers—now teach it through the evidence of the senses in the way I have described.

We can call the proposition ‘Three is two and one’ a definition of the number three, and describe arithmetic (as geometry has been described) as a science based on definitions. But they’re ‘definitions’ in the geometrical sense, not the logical; i.e. they assert not only •the meaning of a term but also •an observed matter of fact. The proposition ‘A circle is a figure bounded by a line which has all its points equally distant from a point within it’ is called the definition of circle; but the proposition from which so many consequences follow—the proposition that really is a first principle in geometry—is that figures answering to this description exist. Similarly, we may call ‘Three is two and one’ a definition of three; but the calculations that depend on that proposition follow not from the definition itself but from an arithmetical theorem presupposed in it, namely that there are collections of objects which while they impress the senses thus:

•••

can be separated into two parts thus:

•

• •

This proposition being granted, we call all such parcels ‘threes’; and then the statement of the above-mentioned physical fact will serve also as a definition of ‘three’.

The science of number is thus no exception to the conclusion I have argued for, that the processes even of deductive sciences are entirely inductive, and that their first principles are generalisations from experience. One last question remains. You’ll remember this finding about geometry:

Some of its inductions are not exactly true; the special certainty ascribed to it, leading men to call its propositions ‘necessary truths’, is fictitious and hypothetical, being true only in the sense that those propositions validly follow from the hypothesis of the truth of premises, which are admittedly mere approximations to truth.

Is that also true of the propositions of the science of number?

§3. The inductions of arithmetic are of two sorts: (i) the likes of ‘One and one are two’, ‘Two and one are three’ etc., which can be called the definitions of the various numbers in the improper or geometrical sense of ‘definition’; and (ii) the two axioms ‘The sums of equals are equal’, ‘The differences of equals are equal’. These two are enough, for the corresponding propositions about unequals can be proved from these by a reductio ad absurdum.

These axioms and the so-called definitions are (I repeat) results of induction; true of all objects whatever, and they may seem to

be exactly true, just as they stand

rather than merely

having the status of exact truths conditionally, i.e. on the condition that a certain assumption is true.

So it is natural to infer that the conclusions of the science of number are exactly true, making it unlike the other demonstrative sciences in having results that are categorically certain, not merely hypothetically so.

When we look more closely, though, we find that even here there is one hypothetical element in the ratiocination. In all propositions about numbers a condition is implied, without which none of them would be true; and that condition is an assumption that may be false. The condition is that 1 = 1, i.e. that all the numbers are numbers of the same or of equal units. [That clause is verbatim from Mill.] If this is doubtful, not one of the propositions of arithmetic will hold true. How can we know that one pound and one pound make two pounds if one of the pounds may be troy and the other avoirdupois?. . . . How can we know that a forty-horse power is always equal to itself, unless we assume that all horses are of equal strength? It’s certain that 1 is always equal in number to 1; and where all that matters is the mere number of objects or object-parts, the conclusions of arithmetic are true without mixture of hypothesis. There are such cases in statistics, e.g. in an inquiry into the size of the population of any country. In that inquiry we don’t care whether they are adults or children, strong or weak, tall or short; all we want to ascertain is their number. But whenever from (in)equality of number we infer (in)equality in any •other respect, the arithmetic we bring to such inquiries becomes as hypothetical a science as geometry. All units must be assumed to be equal in that •other respect; and this is never precisely true because one actual pound weight is not exactly equal to another, nor one measured mile’s length to another; a more exact balance or more accurate measuring instruments would always detect some difference.

What is commonly called ‘mathematical certainty’, therefore, which implies •unconditional truth and •perfect accuracy, is an attribute not of all mathematical truths but only of those that relate to pure number as distinguished from quantity in the more enlarged sense; and only if we abstain from supposing that the numbers are a precise pointer to actual quantities. . . .

§4. Thus, we find that the method of all deductive sciences is hypothetical. They trace the consequences of certain assumptions, leaving the questions

•Are the assumptions true? and

•If not, are they near enough to true?

to be answered later. The reason ·for proceeding in this way· is obvious. Setting aside the special case of propositions that are purely about number and not applied to anything else, we can see that in every other case of deductive investigation we need to determine how far short of exactly true the relevant assumptions are. This is generally a matter of observation, to be repeated in every new case; and if it has to be settled by argument rather than observation, those arguments may vary in length, complexity, and other factors from case to case. But the other part of the process—namely, determining what we can conclude if (and to the extent that) we find the assumptions to be true—can be done once for all, and the results held ready to be used when needed. We are doing beforehand everything that can be done beforehand, so as to minimize the work that has to be done in particular cases that press us for a decision. This inquiry into the inferences that can be drawn from assumptions is what is properly called demonstrative science.

It is of course quite as practicable to arrive at new conclusions from facts assumed, as from facts observed; from fictitious inductions as from real ones. Deduction consists of a series of inferences in this form—a is a mark of b, b of c, c of d, therefore a is a mark of d, which last may be a truth inaccessible to direct observation. Similarly, it is allowable to say ‘Suppose that a were a mark of b, b of c, and c of d, a would be a mark of d’, which last conclusion was not thought of by those who laid down the premises. A system of propositions as complicated as geometry could be deduced from assumptions that are false, as was done by Ptolemy, Descartes, and others, in their attempts to explain. . . .the phenomena of the solar system on the supposition that the apparent motions of the heavenly bodies were the real motions, or were produced in some way different from the actual one. Sometimes the same thing is knowingly done so as to show that the assumption is false; which is called a reductio ad absurdum. Such reasoning goes like this: ‘a is a mark of b, and b of c; now if c were also a mark of d, a would be a mark of d; but d is known to be a mark of the absence of a; consequently a would be a mark of its own absence, which is a contradiction; therefore c is not a mark of d.

§5. Some writers have held that all ratiocination ultimately comes down to reductio ad absurdum; because we can enforce assent to a conclusion P. . . .by showing that if the P is denied we must deny at least one of the premises, and as they are all supposed to be true, that would be a contradiction. In line with this, many people have thought that the special nature of the evidentness of ratiocination consists in the impossibility of admitting the premises and rejecting the conclusion without a contradiction in terms. This theory, however, can’t explain the grounds on which ratiocination itself rests. If someone denies the conclusion despite his admission of the premises, he isn’t involved in any direct and explicit contradiction until he is compelled to deny some premise; and he can only be forced to do this by a reductio ad absurdum, i.e. by another ratiocination. But if he denies the validity of the reasoning process itself, he can’t be forced to assent to the second syllogism any more than he can to the first. Thus, no-one is ever forced to a contradiction in terms: he can only be forced to an infringement of the fundamental maxim of ratiocination, namely that whatever has a mark also has what it is a mark of. . . .

That’s as far as I can go just now in the theory of deduction. Further insight into the subject requires that we lay the foundation of the philosophic theory of induction itself; when we do that, the theory of deduction will automatically fall into place because deduction is, as I have shown, a kind of induction. In that context, deduction will receive its share of whatever light may be thrown on the great intellectual operation of which it forms such an important part.

Chapter 7. Examining some of the opposition to the preceding doctrines

§1. An opinion that stands in need of much illustration can often receive it most effectively and least tediously in the form of a defence against objections. And someone who advances a doctrine on a subject concerning which theorists are still divided has a duty to examine, and to the best of his ability to judge, the opinions of other thinkers. [The first four sections of this chapter are addressed to Herbert Spencer; the fifth to Sir William Hamilton.]

Mr. Herbert Spencer has criticised some of the doctrines of chapters 5 and 6, and propounded a theory of his own on the subject of first principles. He agrees with me in considering axioms to be ‘simply our earliest inductions from experience’. But he differs ‘widely’ from me ‘concerning the worth of the test of inconceivability’. He thinks that it is the ultimate test of all beliefs, a conclusion that he reaches by two steps. (i) We never can have any stronger ground for believing anything than that the belief of it ‘invariably exists’. Whenever any fact or proposition is invariably believed— which I think Spencer means ‘believed by everyone (oneself included) at all times’—it is entitled to be accepted as one of the primitive truths or original premises of our knowledge. (ii) The criterion by which we decide whether something is invariably believed to be true is our inability to conceive it as false. ‘The inconceivability of its negation is the test by which we ascertain whether a given belief invariably exists or not.’ ‘The only reason we can give for our primary beliefs is the fact of their invariable existence, tested by our trying and failing not to have them.’ He thinks that this our only reason for believing in our own sensations. If I believe that I feel cold, I accept this as true only because I can’t conceive that I am not feeling cold. ‘While the proposition remains true, the negation of it remains inconceivable.’ There are numerous other beliefs that Spencer thinks rest on the same basis; most of them being propositions that the metaphysicians of the Reid and Stewart school regard as truths of immediate intuition—

•There exists a material world;

•This is the very world that we directly and immediately perceive, and not merely the hidden cause of our perceptions

•Space, time, force, extension, figure are not modes of our consciousness, but objective realities;

—these are regarded by Spencer as truths known by the inconceivability of their negations. He holds that we can’t by any effort conceive these objects of thought as mere states of our mind—as not existing external to us. So their real existence is as certain as our sensations themselves. According to this doctrine, truths of which we have direct knowledge are known to be truths only by the inconceivability of their negations; and truths of which we don’t have direct knowledge are known to be truths only because they are derived from truths of the first sort; and those derivations are believed to be valid only because we can’t conceive them not to be. So inconceivability is the ultimate ground of all assured beliefs.

Up to here, Spencer’s doctrine doesn’t differ much from the ordinary view of philosophers of the intuitive school, from Descartes to Whewell; but at this point he parts company with them. For he doesn’t follow them in setting up the test of inconceivability as infallible. On the contrary, he holds that the test may be fallacious, not from any fault in the test itself but because ‘men have thought to be inconceivable some things that were not inconceivable’. And he himself denies plenty of propositions that are usually regarded as among the most striking examples of truths whose negations are inconceivable. But all tests, he says, occasionally fail; if such failure undercuts ‘the test of inconceivability’, it ‘must similarly undercut all tests whatever’. He continues:

‘We consider a conclusion logically inferred from established premises to be true. Yet in millions of cases men have been wrong in the inferences they have thought were logical. Should we infer from this fact that it’s absurd to consider a conclusion as true simply because it is logically drawn from established premises? No: we should say this:

Although men may have regarded as logical some inferences that were not logical, there are logical inferences, and until we are better instructed we are justified in assuming the truth of what seem to us to be such.

Similarly, although men may have regarded as inconceivable some things that were not so, there may still be inconceivable things; and our inability to conceive the negation of P may still be our best warrant for believing that P. It may sometimes turn out to be an imperfect test; but it’s the best test we have for our most certain beliefs; so doubting a belief because we have no higher guarantee for it is really doubting all beliefs.’

So Spencer’s doctrine doesn’t erect the curable limitations of the human conceptive faculty into laws of the outward universe; only the incurable limitations.

§2. Spencer has two arguments to support his doctrine that ‘a belief that is proved by the inconceivability of its negation to invariably exist is true’. One is positive, the other negative.

The positive argument says that every such belief represents the aggregate of all past experience. Spencer writes:

‘Conceding the entire truth of the view that •during any phase of human progress what men can specifically conceive depends entirely on the experiences they have had; and that •by widening their experiences they may eventually become able to conceive things that used to be inconceivable to them, it can still be argued that

because at any time the best warrant men can have for a belief is the perfect agreement of all their previous experience in support of it, it follows that at any time the inconceivability of its negation is the deepest test any belief is capable of. . .

Objective facts are always impressing themselves upon us; our experience is a record of these objective facts; and something’s being inconceivable implies that it is wholly at variance with the record. Even if this were the whole story, it’s not clear how, if every truth is primarily inductive, any better test of truth could exist. But it must be remembered that while many of these facts that impress themselves upon us are •occasional, and others are merely •very general, some are •universal and unchanging. These universal and unchanging facts are, by the hypothesis, certain to establish beliefs the negations of which are inconceivable; while the others are not certain to do this; and if they do, subsequent facts will reverse their action. So if after an immense accumulation of experiences there remain beliefs the negations of which are still inconceivable, most and perhaps all of them must correspond to universal objective facts. If

•there are. . . absolute uniformities in nature, if

•these uniformities produce (as they must) absolute uniformities in our experience, and if. . .

•these absolute uniformities in our experience make us unable to conceive their negations, then

•corresponding to each absolute uniformity in nature that we can know, there must exist in us a belief the negation of which is inconceivable, and which is absolutely true.

In this wide range of cases subjective inconceivability must correspond to objective impossibility. Further experience will produce correspondence where it may not yet exist; and we may expect the correspondence to become ultimately complete. In nearly all cases this test of inconceivability must be valid now’

—I wish I could think we were so near to omniscience!—

•‘and where it isn’t, it still expresses the net result of our experience up to the present time, which is the most that any test can do.’

To this I have two answers. (1) It is by no means true that the inconceivability by us of the negative of a proposition proves that any—let alone all—‘previous experience’ has been in favour of the affirmative. There may have been no such previous existing experiences but only a mistaken supposition of them. How did the inconceivability of antipodes prove that experience had given any testimony against their possibility? How did the incapacity men felt of conceiving sunset as anything but a motion of the sun, represent any ‘net result’ of experience in support of its being the sun and not the earth that moves? What is represented is not •experience but only •a superficial semblance of experience. All that is proved with regard to real experience is the negative fact that men have not had experience that would have made the inconceivable proposition conceivable.

(2) Even if it were true that inconceivability represents [Spencer’s word was ‘expresses’] the net result of all past experience, why should we settle for the representative when we can get at the thing represented? If our inability to conceive the negation of P is proof of P’s truth, because it proves that our experience so far has been uniformly in its favour, the real evidence for P is not •the inconceivability but •the uniformity of experience. If all past experience is in favour of P, let this be stated and the belief openly based on it ·without an irrelevant detour through the inconceivability of not-P·. And then we can consider what that fact ·about experience· is worth as evidence of P’s truth. In some cases uniformity of experience is strong evidence, in some it is weak, and in others again it scarcely amounts to evidence at all. That all metals sink in water was a uniform experience, from the origin of the human race to the discovery of potassium in the present century by Sir Humphry Davy. That all swans are white was a uniform experience down to the discovery of ·black swans in· Australia. In the few cases where uniformity of experience does amount to the strongest possible proof, as with propositions such as these,

•Two straight lines can’t enclose a space,

•Every event has a cause,

it’s not because their negations are inconceivable, which is not always the fact, but because the experience that has been uniform in their favour pervades all nature. I’ll shown in Book III that none of the conclusions either of induction or of deduction can be considered certain except as far as their truth is shown to be inseparably bound up with truths of this class.

I maintain then (2) that uniformity of past experience is •far from being a universally sound criterion of truth, and (1) that inconceivability is even •further from being a test of that test. Uniformity of contrary experience is only one of many causes of inconceivability. One of the commonest is tradition handed down from a period of more limited knowledge. The mere familiarity of one way of producing a phenomenon is often enough to make every other way seem inconceivable. Whatever connects two ideas by a strong association may, and continually does, make it impossible to separate them in thought; as Spencer frequently recognises in other parts of his work. Why were the Cartesians unable to conceive that one body could produce motion in another without contact? It wasn’t their lack of relevant experience. They had as much experience of that way of producing motion as they had of other ways. The planets had revolved, and heavy bodies had fallen, every hour of their lives. But they fancied these phenomena to be produced by a hidden machinery that they •didn’t see, because without it they couldn’t conceive what they •did see. The inconceivability, instead of representing their experience, dominated and overrode their experience. I now turn to Spencer’s negative argument, on which he lays more stress.

§3. The negative argument says: whether the inconceivability of not-P is good evidence or bad evidence for P, no stronger evidence can be obtained. That what is inconceivable can’t be true is postulated in every act of thought. It is the foundation of all our original premises, and is assumed still more ·strongly· in all conclusions from those premises. The invariability of belief, tested by the inconceivability of its negation, ‘is our sole warrant for every demonstration. Logic is simply a systematisation of the process by which we •indirectly obtain this warrant for beliefs that don’t •directly possess it. To gain the strongest conviction possible regarding any complex fact, we either •work back through propositions that it comes from, unconsciously testing each by the inconceivability of its negation, until we reach some axiom or truth that we have similarly tested; or we •work forward through propositions that are implied by it, testing each in the same way. In either case we connect some isolated belief with a belief that invariably exists, by a series of intermediate beliefs that invariably exist.’ This sums up the theory:

‘When we perceive that the negation of a belief is inconceivable, we have all possible warrant for asserting the invariability of its existence; and in asserting this we express both •our logical justification for it and •the inexorable necessity we are under of holding it. . . We have seen that this is the assumption on which every conclusion whatever ultimately rests. We have no other guarantee

•for the reality of consciousness, of sensations, of personal existence;

•for any axiom;

•for any step in a demonstration.

Hence, as being taken for granted in every act of the understanding, it must be regarded as the universal postulate.’

But this postulate that we are under an ‘inexorable necessity’ of holding true is sometimes false; ‘beliefs that once were shown by the inconceivability of their negations to exist invariably have since been found untrue’; and ‘beliefs that now have this character may some day share the same fate’; ·and Spencer knows all this, so· the canon of belief he lays down is that ‘the most certain conclusion’ is the one that ‘involves the postulate the fewest times’. So reasoning ought never to prevail against one of the immediate beliefs (the belief in matter, in the outward reality of extension, space, and the like), because each of these involves the postulate only once; while a bit of reasoning involves the postulate in the premises and involves it again at every step of the ratiocination. Why? Because each step in the argument is recognised as valid only because we can’t conceive the conclusion not to follow from the premises.

It will be convenient to take the last part of this argument first. In every reasoning, according to Spencer, the assumption of the postulate is renewed at every step. At each inference we judge that the conclusion follows from the premises, our sole warrant for that judgment being that we can’t conceive it not to follow. Consequently if the postulate is fallible, the conclusions of reasoning are harmed by that uncertainty more than direct intuitions are; and the more steps the argument has the greater the disproportion.

To test this doctrine, let us start with an argument consisting only of a single step, which would be represented by one syllogism. This argument does rest on an assumption, and we have seen what it is, namely that whatever has a mark has what it is a mark of. I shan’t discuss here the basis for this axiom; let us suppose (with Spencer) that it rests on the inconceivability of its reverse.

Let us now add a second step to the argument: we require. . . what? Another assumption? No—the same assumption a second time; and so on to a third, and a fourth. I don’t see how, on Spencer’s own principles, the repetition of the assumption weakens the argument’s force. If the second step required us to assume some other axiom, the argument would be weakened, because it would now run two risks of falsity instead of only one. But in fact only one axiom is required, and if it is true once it is true every time; if a 100-step argument assumed the axiom a hundred times, these hundred assumptions would create only one chance of error altogether. On Spencer’s theory the deductions of pure mathematics are among the most uncertain of argumentative processes, because they are the longest. But the number of steps in an argument does not subtract from its reliableness, if no new premises of an uncertain character are taken up along the way. [Mill here has a long footnote stating and replying to two arguments that Spencer presented against what Mill has been saying in this section. The first argument is flatly wrong, while the second is at best marginal. There is nothing much to be learned from this exchange.]

Now for the premises. Spencer holds that our assurance of their truth—whether they are generalities or individual facts—is based on the inconceivability of their being false. Now, the word ‘inconceivable’ is ambiguous; Spencer is aware of this ambiguity and would sincerely deny that he is founding an argument on it; but it is in fact at work helping him to make his case. ‘Inconceivability’ sometimes means inability to form or get rid of an •idea; sometimes inability to form or get rid of a •belief. The former meaning is the better one because a ‘conception’ is always an idea and never a belief. But in philosophical discussion the word is used with its wrong meaning at least as often as with the right one; and the intuitive school of metaphysicians needs both. To see the difference, consider these two contrasted examples. (a) The early scientists considered antipodes incredible because ‘inconceivable’. But antipodes weren’t inconceivable in the original sense of the word. An idea of them could be formed without difficulty: they could be completely pictured to the mental eye. What was difficult—and to them seemed impossible—was to find them believable. They could assemble the idea of men sticking on by their feet to the under side of the earth; but the belief would follow that they must fall off. Antipodes were not unimaginable, but they were unbelievable.

(b) On the other hand, when I try to conceive an end to space, the two ideas refuse to come together. When I try to form a conception of the last point of space, I can’t help picturing to myself a vast space beyond that last point. The combination is, under the conditions of our experience, unimaginable. It’s important to bear in mind this double meaning of ‘inconceivable’, because the argument from inconceivability almost always depends on switching between those two meanings.

When Spencer tests the truth of a proposition by asking whether its negation is ‘inconceivable’, which of the two senses is he giving to that word? I inferred from the course of his argument that he meant ‘unbelievable’; but he has recently disclaimed this meaning and declared that by an ‘inconceivable’ proposition he always means ‘a proposition the terms of which can’t by any effort be brought before consciousness in the relation that the proposition asserts between them—a proposition the subject and predicate of which offer an insurmountable resistance to union in thought’. So now we know that Spencer always •tries to use ‘inconceivable’ in its proper sense; but there’s evidence that he doesn’t always •succeed, and that the other sense of the word—the popular sense—sometimes creeps in with its associations and prevents him from clearly separating the two. For example, when he says that when I feel cold I can’t conceive that I’m not feeling cold, he can’t mean ‘I can’t conceive myself not feeling cold’, for it’s obvious that I can. In this context, therefore, ‘conceive’ is being used to express the recognition of a matter of fact—the perception of truth or falsehood; and I take this to be about belief as distinguished from simple conception. Again, Spencer calls the attempt to conceive something that is inconceivable ‘an abortive effort to cause the non-existence. . . ’—not of a conception or mental representation but of a belief. So we need to revise a considerable part of what Spencer writes, if it is to be kept consistent with his definition of ‘inconceivability’.

Mill’s next sentence: But in truth the point is of little importance, since inconceivability in Spencer’s theory is only a test of truth, inasmuch as it is a test of believability.

what he should have said, given what follows: But in fact we can’t amend what he says so that it consistently uses ‘inconceivable’ in its proper sense; because in his theory inconceivability is a test of truth only because it is a test of believability; which means that the improper sense of ‘inconceivable’ has a structural role in his theory.

The inconceivability of P is the extreme case of P’s unbelievability. This is the very foundation of Spencer’s doctrine. For him the invariableness of the belief is the real guarantee. The attempt to •conceive the negative is made so as to test the inevitableness of the •belief, so it should be called an attempt to •believe the negative. When Spencer says that while looking at the sun a man can’t conceive that he is looking into darkness, he should have said that the man can’t believe that he is doing so. For it is surely possible in broad daylight to imagine oneself looking into darkness.[1] As Spencer himself says, speaking of the belief in our own existence, ‘He can conceive well enough that he might not exist, but that he does not exist he finds it impossible to conceive’, i.e. to believe. So his statement comes down to this: ‘I believe that I exist and that I have sensations, because I can’t believe otherwise.’ And in this case everyone will agree that the impossibility is real. Each person inevitably believes in his present sensations or other states of subjective consciousness. They are facts known through themselves ·and not as conclusions inferred from premises·; it is impossible to ascend beyond them [= ‘to find any source or origin for them’]. Their negation is really unbelievable, so there’s never any question of believing it. Spencer’s theory is not needed for these truths.

But according to him there are other beliefs, relating to things other than our own subjective feelings, for which we have the same guarantee—which are similarly invariable and necessary. These other beliefs can’t be necessary because they don’t always exist. There have been and still are many people who don’t believe in the reality of an external world, let alone the reality of extension and shape as the forms of that external world; who don’t believe that space and time exist independently of the mind—or any other of Spencer’s objective intuitions. The negations of these allegedly invariable beliefs are not unbelievable, for they are believed! It isn’t obviously wrong to say that we can’t imagine tangible objects as mere states of our own and other people’s consciousness; that the perception of them irresistibly suggests to us the idea of something external to ourselves: and I’m not in a position to say that this is not the fact (though I don’t think anyone is entitled to affirm it of anyone else). But many thinkers have •believed—whether or not they could •conceive it—that what we represent to ourselves as material objects are mere states of consciousness, complex feelings of touch and of muscular action. Spencer may think the inference from the unimaginable to the unbelievable is correct because he holds that belief is merely the persistence of an idea, so that what we can succeed in imagining we can’t at that moment help finding believable. But what does it matter what we find at the moment if the moment is in contradiction to the permanent state of our mind? A man who was as an infant frightened by stories of ghosts, though he disbelieves them in later years (and perhaps never believed them), may throughout the rest of his life be disturbed by being in a dark place in circumstances that stimulate his imagination. The idea of ghosts, with all its attendant terrors, is irresistibly called up in his mind by the outward circumstances. Spencer may say that while the man is under the influence of this terror he has a temporary and uncontrollable belief in ghosts. Be it so; but if that is how things stand, what is the truest to say about this man on the whole—that he believes in ghosts, or that he doesn’t believe in them? Assuredly that he doesn’t believe in them. It’s like that with those who disbelieve a material world. Though they can’t get rid of the idea; though while looking at a solid object they can’t help having the conception of (and therefore, according to Spencer’s metaphysics, the momentary belief in) its externality; even at that moment they would sincerely deny holding that belief; and it would be incorrect to call them anything but disbelievers of the doctrine. So the belief is not invariable; and the ‘inconceivability’ test for whether someone has the belief fails.

For a familiar illustration of the fact that it’s perfectly possible to •believe something without finding it •conceivable. . . ., consider an educated person’s state of mind regarding sunrise and sunset. Every educated person knows by investigation or believes on the authority of science that it’s the earth and not the sun that moves: but there are probably few who habitually conceive this phenomenon as anything but the ascent or descent of the sun. Certainly no-one can do this without working at it for a long time; and it’s probably no easier now than in the first generation after Copernicus. Spencer does not say ‘In looking at sunrise it’s impossible not to conceive that it is the sun that moves, therefore this is what everybody believes, and we have all the evidence for it that we can have for any truth’. Yet this would be an exact parallel to what he says about the belief in matter. [The conceptual feat that Mill here describes as difficult is indeed so. For help in performing the feat, see the famous passage on pp. 30–34 of Paul Churchland’s Scientific Realism and the Plasticity of Mind (Cambridge U.P., 1979).]

The existence of matter and other noumena [see Glossary], as distinct from the phenomenal world, has long been and still is a matter for debate; and the very general belief in them—though it’s not necessary or universal—stands as a psychological phenomenon to be explained, either on the hypothesis of its truth or on some other. The belief isn’t a conclusive proof of its own truth. . . .; but it’s a fact that challenges antagonists to show how such a general and apparently spontaneous belief can have originated if not from the real existence of the thing believed. And its opponents have never hesitated to accept this challenge.[2] How much success they have in meeting it will probably determine the ultimate verdict of philosophers on the question.

§4. In a recent writing Spencer resumes what he rightly calls the ‘friendly controversy that has been long pending between us’; and expresses his regret (which I cordially share) that ‘this lengthened exposition of a single point of difference, not accompanied by an exposition of our many points of agreement, inevitably produces an appearance of much more disagreement than actually exists’. I agree with Spencer that the difference between us, if measured by our conclusions, is ‘superficial rather than substantial’; and I greatly value being, in the field of analytic psychology, in so much agreement with a thinker of his force and depth. But I also agree with him that the difference between his premises and mine has ‘profound importance, philosophically considered’; and neither of us should walk out on it until the whole case for each side has been fully examined and discussed.

In his latest statement of the universal postulate Spencer has replaced ‘beliefs that invariably exist’ by ‘cognitions [see Glossary] of which the predicates invariably exist along with their subjects’. And he argues like this:

(i) A failed attempt to conceive the negation of a proposition shows that the cognition it expresses is one the predicate of which invariably exists along with its subject; and

(ii) The discovery that the predicate invariably exists along with its subject is the discovery that this cognition is one we are compelled to accept.

Therefore

(iii) A failed attempt to conceive the negation of a proposition P shows that we are compelled to accept P.

I accept both premises of Spencer’s syllogism, but in different senses of the middle term (·‘the predicate of which invariably exists along with its subject’·. If this is understood in its most obvious meaning, as an existence in actual nature— i.e. in our objective sensation-involving experience—I do of course admit (ii) that when we have ascertained this we are compelled to accept the proposition: but then I don’t admit (i) that the failure of an attempt to conceive the negation proves the predicate to be always co-existent with the subject in actual nature. But I think that Spencer intends his middle term in its other sense, in which ‘the invariable existence of the predicate along with the subject’ means only that the one is inseparable from the other in our thoughts; then indeed (i) the inability to separate the two ideas proves their inseparable conjunction, here and now, in the mind that has failed in the attempt; but this inseparability in thought does not prove (ii) a corresponding inseparability of subject and predicate in fact—or even in the thoughts of other people, or of the same person in a possible future.

‘That some propositions have been wrongly accepted as true, because their negations were thought to be inconceivable when they were not’, does not, in Spencer’s opinion, ‘disprove the validity of the test’. He gives two reasons why. (a) Any test ‘is liable to yield untrue results because of stupidity or carelessness in those who use it.’, (b) The propositions in question ‘were complex propositions, not to be established by a test that is valid only for propositions that have no analysable complexity’. ‘A test that is legitimate for a simple proposition whose subject and predicate are directly related is not legitimate for a complex proposition whose subject and predicate are indirectly related through the many simple propositions implied.’ ‘That things which are equal to the same thing are equal to one another is a fact that can be known by direct comparison of actual or ideal relations. . . But that the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides can’t be known immediately by comparison of two states of consciousness: here the truth can be reached only through a series of simple judgments concerning the (un)likenesses of certain relations.’. . . .

It’s only fair to give Spencer’s doctrine the benefit of the limitation he claims—namely that it is applicable only to propositions that are assented to on simple inspection, without any need for proof. But this limitation doesn’t exclude some of the most conspicuous examples of propositions that are now known to be false or groundless but whose negations were once found to be inconceivable—such as the proposition that in sunrise and sunset it is the sun that moves, that gravitation may exist without an intervening medium, and even the case of antipodes. . . . When consciousness is confronted by one of Spencer’s ‘complex’ propositions, without an accompanying proof, it gives no verdict at all: it doesn’t find it inconceivable •that the square of the hypotenuse equals the sum of the squares of the other two sides, or •that it doesn’t equal that sum. But in the three cases that I have just cited, the inconceivability seems to be found directly; no train of argument was needed to obtain the verdict of consciousness on the point. . . . They are cases where one of two opposite predicates seemed immediately ·and intuitively· to be incompatible with the subject, and the other therefore to be proved always to exist with it.[3]

As now limited by Spencer, the ultimate cognitions fit to undergo his test are only ones that are so universal and elementary that they are represented in the earliest and most unvarying experience (or apparent experience) of all mankind. If in such a case the negation really is inconceivable, that is explained by the experience. And I have asked: Why should the truth be tested by inconceivability, when we can go further back for proof—namely to the experience itself? Spencer replies that the experiences can’t be all recalled to mind, and if they could be recalled there would be too many of them for us to manage. He seems to understand ‘test a proposition by experience’ as meaning that ‘before accepting as certain the proposition that any rectilineal figure must have as many angles as it has sides’ I have ‘to think of every triangle, square, pentagon, hexagon etc. that I have ever seen, and to verify the asserted relation in each case.’ I can only say, with surprise, that I don’t take this to be the meaning of ‘appeal to experience’. It is enough to know that one has been seeing the fact all one’s life without ever noticing any instance to the contrary, and that other people. . . .unanimously declare the same thing. . . . These remarks don’t lose their force even if we believe as Spencer does that mental tendencies originally derived from experience impress themselves permanently on the cerebral structure and are transmitted by inheritance, so that modes of thinking that are acquired by the race become innate and a priori in the individual, representing the experience of his ancestors in addition to his own. All that would follow from this is that a conviction might be really innate—i.e. prior to individual experience—and yet not be true, because the inherited tendency to accept it may have originally been caused by something other than its truth. . . .

§5. Sir William Hamilton holds, as I do, that inconceivability is no criterion of impossibility. ‘There is no ground for inferring a certain fact to be impossible merely from our inability to conceive its possibility.’ ‘There are things that may—indeed must—be true though the understanding is wholly unable to construe to itself the possibility.’ But Hamilton is a firm believer in the a priori character of many axioms and of the sciences deduced from them; and he’s so far from basing those axioms on experience that he declares some of them to be true even of noumena—of ‘the unconditioned’—which he says our faculties can’t give us any knowledge of. He credits two axioms with this exceptional emancipation from the limits that confine all our other possibilities of knowledge—two chinks through which a ray of light finds its way to us from behind the curtain that veils from us the mysterious world of ‘things in themselves’. He follows the scholastics in naming them:

•the principle of contradiction, which says that two contradictory propositions can’t both be true; and

•the principle of excluded middle, which says that two contradictory propositions can’t both be false.

Armed with these logical weapons, we can boldly face ‘things in themselves’ and confront them with a choice, knowing for sure that they absolutely must choose one side or the other, though we may be never be able to discover which. To take his favourite example, we can’t conceive •the infinite divisibility of matter, and we can’t conceive a minimum, i.e. an end to divisibility; yet one or the other must be true.

As I haven’t yet said anything about those the two axioms, this is not a bad place to consider them. The former asserts that an affirmative proposition and the corresponding negative proposition can’t both be true; which has generally been held to be intuitively evident. Hamilton and the Germans see it as a statement in words of a law of our thinking faculty. Other equally considerable philosophers deem it to be an identical proposition [see Glossary], an assertion that comes from the meaning of terms, a way of defining ‘negation’ and ‘not’.

I can go one step with the latter group. An affirmative assertion and its negation are not two independent assertions connected with each other only by the relation of as mutually incompatibility. If the negation is true the affirmative must be false really is a mere identical proposition; for the negation asserts nothing but the falsity of the other. So we should drop the ambitious phraseology that gives the principium contradictionis the air of a fundamental antithesis pervading nature, and express it in the simpler form A proposition can’t be false and true at the same time. But I can’t follow the nominalists in taking the further step of declaring this to be a merely verbal proposition. I think it is like other axioms in being one of our first and most familiar generalisations from experience. The ultimate foundation of it I take to be that belief and disbelief are two different mental states, excluding one another. We know this by the simplest observation of our own minds. And if we carry our observation outward, we also find that

•light and darkness,

•sound and silence,

•motion and stillness,

•equality and inequality,

•preceding and following,

•successiveness and simultaneity,

any positive phenomenon whatever and its negation, are distinct phenomena, pointedly contrasted, and the one always absent where the other is present. I consider the principle of contradiction to be a generalisation from all these facts.

The principle of excluded middle (or that one of two contradictories must be true) means that an assertion must be either true or false: either the affirmative is true or otherwise its negation is true in which case the affirmative is false. It’s surprising to have this principle described as a so-called necessity of thought, because it isn’t even true just as it stands. A proposition must be either true or false if the predicate can in any intelligible sense be attributed to the subject; it’s because this is always assumed to be the case in treatises on logic that the axiom is always presented as absolutely ·rather than merely hypothetically· true. ‘Abracadabra is a second intention’ is neither true nor false. Between •true and •false there’s a third possibility, •meaningless: and this alternative is fatal to Hamilton’s extension of the maxim to noumena. ‘Matter must either have a minimum of divisibility or be infinitely divisible’—that’s more than we can ever know. (a) Matter, in any but the phenomenal sense of the term, may not exist; and it will hardly be said that a nonentity must be either infinitely or finitely divisible! (a) And although matter, considered as the hidden cause of our sensations, really does exist, what we call ‘divisibility’ may be an attribute only of our sensations of sight and touch and not of their unknowable cause. Perhaps it doesn’t make sense to predicate divisibility of things in themselves, or therefore of matter in itself; in which case it isn’t true that matter in itself must be either infinitely or finitely divisible.

I’m glad to be in complete agreement on this question with Herbert Spencer. I now quote a paragraph from a recent paper of his; the germ of an idea identical with his may be in what I have written here, but in Spencer it is not an undeveloped thought but a philosophical theory.

‘When remembering a certain thing as in a certain place, the place and the thing are mentally represented together; while to think of the non-existence of the thing in that place implies a consciousness in which the place is represented but the thing isn’t. Similarly, if instead of thinking of an object as colourless we think of its having colour, the change consists in the addition to the concept of an element that was absent from it before—the object can’t be thought of first as red and then as not red, without one component of the thought being totally expelled from the mind by another. The law of the excluded middle, then, is simply a generalisation of the universal experience that some mental states are directly destructive of other states. It formulates a certain absolutely constant law, that the appearance of any positive mode of consciousness can’t occur without excluding a corresponding negative mode; and that the negative mode can’t occur without excluding the corresponding positive mode—the positive/negative antithesis being merely an expression of this experience. Hence it follows that if consciousness is not in one of the two modes it must be in the other.’

I must now close this supplementary chapter, and with it Book II. The theory of induction, in the most comprehensive sense of the term, will be the subject of Book III.

NOTES

[1] Spencer distinguishes ‘conceiving myself looking into darkness’ from ‘conceiving that I am looking into darkness’. This switch from ‘myself. . . ’ to ‘that I am. . . ’ marks the transition from conception to belief. The form ‘to conceive that P’ is not consistent with using ‘conceive’ in its proper sense.

[2] I have myself accepted the contest, and fought it out on this battle-ground, in my Examination of Sir William Hamilton’s Philosophy, chapter 11.

[3] In one of the cases Spencer surprisingly thinks that the belief of mankind ‘cannot rightly be said to have undergone’ the change I allege. Spencer himself still thinks we can’t conceive gravitation acting through empty space. ‘If an astronomer avowed that he could conceive gravitative force as exercised through space absolutely void, my private opinion would be that he mistook the nature of conception. Conception implies representation. Here the elements of the representation are the two bodies and an agency by which either affects the other. To conceive this agency is to represent it in some terms derived from our experience—i.e. from our sensations. As this agency gives us no sensations, we are obliged (if we try to conceive it) to use symbols idealized from our sensations—imponderable units forming a medium.’ [That sentence is verbatim from Spencer.]

If Spencer means that the action of gravitation gives us no sensations, that’s one of the most startling things I have ever heard from a philosopher. We have the sensation of one body moving toward another—What more do we need? The ‘elements of the representation’ are not two bodies and an ‘agency’ but two bodies and an effect, namely the fact of their approaching one another. . . .

Book III: Induction

Glossary

Chapter 1. Preliminary remarks about induction in general

Chapter 2. Inductions improperly so called

Chapter 3. The ground of induction

Chapter 4. Laws of nature

Chapter 5. The law of universal causation

Chapter 6. The composition of causes

Chapter 7. Observation and experiment

Chapter 8. The four methods of experimental inquiry

Chapter 9. Examples of the four methods

Chapter 10. Plurality of causes, and the intermixture of effects

Chapter 11. The deductive method

Chapter 12. Explaining laws of nature

Chapter 13. Examples of the explanation of laws of nature

Chapter 14. The limits to the explanation of laws of nature. Hypotheses

Chapter 15. Progressive effects. The continued action of causes

Chapter 16. Empirical laws

Chapter 17. Chance and its elimination

Chapter 18. The calculation of chances

Chapter 19. Extending derivative laws to adjacent cases

Chapter 20. Analogy

Chapter 21. Evidence for the law of universal causation

Chapter 22. Uniformities of coexistence that don’t depend on causation

Chapter 23. Approximate generalisations. Probable evidence

Chapter 24. The remaining laws of nature

Chapter 25. The grounds of disbelief

Glossary

art: In this work, ‘art’ is a vehicle for several related ideas: rules, skill, techniques.

circumstances: 1 In Mill’s usage, the ‘circumstances’ of a given experiment are all the details of what is the case when the experiment is performed—not only in the environment but also in the experiment itself.

coextensive: ‘Law L is coextensive with field F’ means not merely that nothing in F is a counter-example to L but that everything in F is an example of L.

coincidence: In Mill’s usage, the coincidence of two events is simply their occurring at the same time (and usually, perhaps, in the same place). What you and I call a ‘coincidence’ is the occurring together of two events that have no causal relation to one another; in Mill’s terminology that is a ‘casual coincidence’. (Be alert to the difference between ‘casual’ and ‘causal’, both of which occur often in this work.) Here he introduces a different sense of ‘coincidence’, which he explains there.

collocation: Arrangement in space; structure. When in the footnote here Mill explains that by ‘the constitution of things’ he means ‘ultimate laws of nature’ and not ‘collocations’, what he is rejecting is the use of ‘constitution’ to mean ‘how things are arranged, structured, in space’. Here we learn that items entering into a ‘collocation’ can include powers = forces as well as physical things.

concomitant: ‘Of a quality, circumstance, etc.: occurring along with something else, accompanying’ (OED).

concurrence: The concurrence of several events is their occurring together, usually meaning at the same time and in roughly the same place. From Latin meaning ‘run together’.

connote: To say that word W connotes attribute A is to say that the meaning of W is such that it can’t apply to anything that doesn’t have A. For example, ‘man’ connotes humanity.

corpuscle: An extremely small bit of matter—far too small for us to be able to pick it out visually. Adjective corpuscular.

cultivation: Carefully developing (a skill or habit), analogous to cultivating roses or cabbages. Here the two are linked metaphorically.

data: Until about the middle of the 20th century ‘data’ was the plural of ‘datum’. Since then it has become a singular mass term, like ‘soup’.

deus ex machina: Latin literally meaning ‘a god out of a machine’, referring to the use of theatrical machinery to float a god onto the stage to make everything come right at a crucial point in a drama. Nearly always the phrase is used metaphorically, to refer to some problem-‘solving’ item that a theorist introduces in a suspiciously convenient way and without good reasons. Here Mill uses the phrase in both ways at once: the suspiciously convenient item that he refers to is literally God.

efficient cause: This is an Aristotelian technical term. The •formal cause of a coin is its design, the plan according to which it was made; its •material cause is the stuff it is made of; its •final cause is its purpose, namely to be used in commerce; and its •efficient cause is the action of the die in stamping the coin out of a metal sheet. So the efficient cause is what you and I would call, simply, ‘the cause’. But here and thereafter Mill is clearly using ‘efficient cause’ to mean something like: real, metaphysically deep, empirically inaccessible causes, as distinct from the mere orderly event-followings that are the only causes Mill believes in.

fact: In Mill’s usage a ‘fact’ can be a state of affairs or an event or a proposition (not necessarily true) asserting the existence of a state of affairs or event. In the present version, no attempt is made to sort all this out.

inductio per enumerationem simplicem: Latin meaning ‘induction by simple enumeration’. This comes from Bacon, who used it meaning something like ‘reaching a generalisation by simply looking at positive instances and naively failing to look for counter-instances or complications’. Mill seems to be using it that way too.

irritability: Proneness to respond to physical stimuli.

luminiferous ether: The ether was a supposed finely divided or gaseous matter pervading the whole universe; ‘luminiferous’ means ‘light-bearing’: it was thought that light consisted of some kind of disturbance of the ether.

material: The ‘material circumstances’ are the circumstances or details that matter. A ‘material change’ is a change that makes a significant difference.

mutatis mutandis: A Latin phrase that is still in current use. It means ‘(mutatis) with changes made (mutandis) in the things that need to be changed’. The use of it implies that it’s obvious what the needed changes are.

natural theology: This is theology based on facts about the natural world, e.g. empirical evidence about what the ‘purposes’ are of parts of organisms etc. In this context, ‘natural’ is the antonym of ‘revealed’.

numeral: A name of a number, usually confined to names like ‘7’ and not like ‘seven’. Mill doesn’t use the word here, but this version uses it instead of ‘name’ in some contexts where the topic is obviously names of numbers.

occult: It means ‘hidden’, but in the early modern period it always carried the extra sense of ‘mysterious, out of reach of ordinary understanding’ or the like. The statement that gravity is an ‘occult force’ meant that the ultimate truth about gravity, whatever it is, won’t be a part of ordinary physics.

original: Sometimes Mill uses this to mean ‘basic’ or ‘foundational’. An ‘original natural agent’ (here) is a natural cause that wasn’t caused by anything we know about. Mill also uses ‘primeval’ and ‘primitive’ with the same meaning.

patient: The same Latin words lie behind three contrasts:

•adjectives: ‘active’ and ‘passive’

•abstract nouns: ‘action’ and ‘passion’

•concrete nouns: ‘agent’ and ‘patient’

We don’t now use ‘passion’ to refer to any undergoing or being-acted-on, or ‘patient’ to refer to anything that is acted on; but until the end of the 19th century both of those uses were current.

petitio principii: A Latin phrase referring to the procedure of offering a ‘proof’ of P from premises that include P. The English name for this used to be ‘begging the question’, but that phrase has recently come to mean ‘raising the question’

popular: It means ‘of the people’; in early modern times it usually doesn’t mean ‘liked by the people’.

precession of the equinoxes: The slow, steady change in the earth’s axis of rotation.

principle: In the passage by Whewell here, the phrase ‘principle of connection’ may mean ‘something that physically connects them’, thus using ‘principle’ in a sense— now obsolete but extremely common in the early modern period—in which it means ‘source’, ‘cause’, ‘driver’, ‘energizer’, or the like. It is certainly used in that sense by Mill here and here and by Powell here.

putrefy: rot; and the rotten state is putrefaction.

quadrature of the cycloid: A cycloid is the curve traced by a point on the rim of a circular wheel rolling on a plane surface. That curve and the line of the surface enclose an area; its quadrature is the process of discovering the size of that area.

resolve: To resolve x into y and z is to analyse x in terms of y and z, to show that all there is to x is y and z, or the like. Mill explains this here. The noun is resolution.

rigor mortis: Latin for ‘stiffness (or rigidity) of death’. Mill calls it ‘cadaveric rigidity’, but these days the Latin phrase is also the colloquial English one.

sagacity: Here it means something like ‘alert intelligence’.

sui generis: Latin for ‘of its own kind’—not significantly like anything else.

synchronous: Occurring at the same time.

type: ‘the real type of scientific induction’ (here) means ‘the central defining paradigm of scientific induction’. Similarly with ‘the type of uncertainty and caprice’ here and ‘the type of a deductive science’ here.

vera causa: Latin meaning ‘true cause’. A technical term of Newton’s. To say that x is a vera causa of y is to say that x is already known about independently of its causing of y, or perhaps (see here) that x could be known about independently etc.

virtue: power, causal capacity, or the like.

vortex: Descartes’s term for a rapidly rotating collection of fine particles. The plural is vortices.

Chapter 1. Preliminary remarks about induction in general

§1. We are now approaching what can be regarded as the principal topic in this work—because it is more intricate than any of the others, and because it concerns a process that I have shown in Book II to be the one that the investigation of nature essentially consists in. I showed that all inference, and consequently all proof and all discovery of truths that aren’t self-evident, consists of inductions and the interpretation of inductions—i.e. that all our knowledge that isn’t intuitive comes from that source. So it has to be accepted that the main question of the science of logic—the question that includes all others—is

What is induction? and what conditions make it legitimate?

Yet professed writers on logic have almost entirely ignored this question. Metaphysicians haven’t altogether neglected its broad outlines. But they haven’t known enough about the processes by which science has actually succeeded in establishing general truths, so that their analysis of the inductive operation, even when perfectly correct, hasn’t been specific enough to be made the foundation of practical rules that could serve •induction itself in the way the rules of the syllogism serve •the interpretation of induction. As for those who have brought physical science to its present state of improvement, never until very recently have they tried seriously to philosophise on the subject; they haven’t regarded their way of arriving at their conclusions—-as distinct from the conclusions themselves—as worth studying. ·It’s a pity, because· all they needed to do to get a complete theory of the process was to focus on the methods that they had been using, and to generalise these and adapt them to all sorts of problems,

§2. For the purposes of the present inquiry, induction can be defined as the operation of discovering and proving general propositions. As I have already shown, the process of indirectly ascertaining individual facts. . . .is a form of the very same process, because (a) general facts are merely collections of particular facts, definite in kind but indefinite in number; and (b) whenever the empirical evidence justifies us in drawing a conclusion about even one unknown case, it would also justify us in drawing a similar conclusion regarding a whole class of cases. The inference either •doesn’t hold at all or •holds in all cases of a certain description—all cases which, in certain definable respects, resemble those we have observed.

If I’m right in maintaining that the principles and rules of inference are the same whether we are inferring general propositions or individual facts, then a complete logic of •the sciences would also be a complete logic of •practical affairs and common life. An analysis of the process by which general truths are arrived at is virtually an analysis of all induction whatever. Why? Because in any legitimate inference from experience, the conclusion could legitimately be a general proposition. Whether we’re inquiring into a scientific principle or an individual fact, and whether we proceed by experiment or by ratiocination, every link in the chain of inferences is essentially inductive, and the legitimacy of the induction depends in both cases on the same conditions.

When a practical inquirer (e.g. an advocate or judge) is trying to ascertain facts for the purposes not of science but of practical affairs, the principles of induction won’t help him with his chief difficulty. It lies not in making his inductions, but in the selection of them—choosing from among all general propositions ascertained to be true the ones that provide marks by which he can trace whether the given subject of study does or doesn’t have the predicate in question. When an advocate is arguing a doubtful question of fact before a jury, the general propositions he appeals to are mostly in themselves pretty trite, and are assented to as soon as stated; his skill lies in bringing his case under those propositions; in calling to mind any known or accepted maxims of probability that can be applied to the case in hand, and selecting from among them those that are most favourable to the case he is trying to make. His success will depend on his natural or acquired sagacity [see Glossary], aided by his knowledge of the particular subject and of subjects allied with it. Invention, though it can be cultivated [see Glossary], can’t be reduced to rule; there’s no science to enable a man to bring to mind what he needs when he needs it.

But when he has thought of something x, science can tell him whether x will suit his purpose or not. When the inquirer or arguer is selecting the inductions out of which he will construct his argument, his only guide is his own knowledge and sagacity. But the validity of the argument he constructs depends on principles, and must be subjected to tests that are the same for all kinds of inquiries—whether the result is •to give someone an estate or •to enrich science with a new general truth. Either way,

(1) The individual facts must be decided on the basis of the senses, or testimony;

(2) The rules of the syllogism will determine whether the case really falls within the formulae of the inductions under which it has been successively brought; and

(3) The legitimacy of the inductions themselves must be decided by other rules. . .

. . . and these rules are what I intend now to investigate. In many everyday practical contexts this third part of the operation is its least difficult part; but we’ve seen that this is also the case in some big scientific fields. I’m referring to the sciences that are principally deductive, especially mathematics, where •the inductions are few in number and so obvious and elementary that they seem not to need any backing from experience, whereas •combining them so as to prove a given theorem or solve a problem may require the utmost powers of invention and contrivance that our species is gifted with.

If you want further confirmation of my claim that the logical processes that •prove particular facts are the very ones that •establish general scientific truths, consider this: In many branches of science there’s a need to prove single facts; they’re as completely individual as any that are debated in a court of justice, but are proved in the same way as the other truths of the science—without lessening in the slightest the homogeneity of its method. Astronomy provides remarkable example of this. Most of the individual facts on which that science bases its most important deductions—

•the sizes of the bodies of the solar system,

•their distances from one another,

•the shape of the earth, and

•the earth’s rotation

—can’t be established by direct observation; they are proved indirectly, using inductions based on other facts that we can more easily reach. [Mill cites the example of the discovery of the moon’s distance from the earth; two direct observations (of the moon’s relation to two widely separated points on the earth’s surface), followed by sheer trigonometry.]

The process by which that individual astronomical fact was ascertained is exactly like those by which astronomy establishes its general truths; and indeed (as I have shown for all legitimate reasoning) a general proposition could have been concluded instead of a single fact. Strictly speaking, indeed, the result of the reasoning is a general proposition; it’s a theorem about the distance from the earth of any inaccessible object, showing how that distance relates to certain other quantities. The moon is almost the only body whose distance from the earth can really be ascertained in this way, but that’s a mere upshot of facts about the other heavenly bodies that make them incapable of providing such data as the application of the theorem requires. The theorem itself is as true of them as it is of the moon. [Mill has here a footnote responding to criticisms by Whewell of Mill’s use of ‘induction’. He says that Whewell’s preferred sense of the word isn’t justifiable by any philosophical arguments] or supported by usage, at least from the time of Reid and Stewart, who are the principal legislators (as far as the English language is concerned) of modern metaphysical terminology.

So we shan’t fall into error if in treating of induction we limit our attention to the establishment of general propositions. The principles and rules of •induction as directed to this end are the principles and rules of •all induction; and the logic of science is the universal logic, applicable to all inquiries in which man can engage.

Chapter 2. Inductions improperly so called

§1. Induction, then, is the mental operation by which we infer that what we know to be true in a particular case or cases will be true in all cases that resemble the former in certain assignable respects. In other words, induction is the process by which we conclude that what is true of certain individuals in a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times.

This definition excludes from the meaning of ‘induction’ various logical operations that are quite often regarded as examples of ‘induction’.

Induction, as I have defined it, is a process of inference from the known to the unknown; so it excludes any process in which the apparent conclusion is no wider than the premises it is drawn from. Yet the common books of logic present something of this latter kind as the most perfect— indeed the only entirely perfect—form of induction! In those books, every process that sets out from a less general and terminates in a more general expression—which admits of being stated in the form ‘This and that A are B, therefore every A is B’—is called an induction, whether or not anything is really concluded in it. And the induction is said not to be perfect unless every single individual of the class A is included in the premise, I.e. unless what we affirm of the class has already been ascertained to be true of every individual in it, so that the supposed conclusion is really a mere re-assertion of the premises. If we say ‘All the planets shine by the sun’s light’ because we have observed that Mercury, Venus, etc. shine by the sun’s light; or that ‘All the Apostles were Jews’ because we know this regarding Peter, Paul, John, and every other apostle—these and their like are called perfect (and the only perfect) inductions. But this is totally different in kind from my kind of induction; it’s not an inference from known facts to unknown facts, but a mere short-hand record of known facts. Their ‘conclusions’ are not really general propositions. In a general proposition the predicate is affirmed or denied of an unlimited number of individuals, namely all that have the properties connoted by the subject of the proposition—all, existing or possible, whether few or many. ‘All men are mortal’ doesn’t mean •all now living but •all men past, present and future. When the word’s signification is limited so as to make it a name only for each of a number n of individuals, designated as such and (as it were) counted off individually, the proposition, despite its general language (‘·All the planets·. . . ’, ‘·All the Apostles·. . . ’) is not a general proposition but merely n singular propositions, written in an abridged form. The operation may be useful, as most forms of abridged notation are; but it’s not a part of the investigation of truth, though it often has an important role in preparing the materials for that investigation.

Just as we can sum up n singular propositions in one proposition that will be apparently—but not really—general, so also we can sum up n general propositions in one proposition that will be apparently—but not really—more general. Suppose that for each distinct species of animals we establish by induction that every animal of that species has a nervous system, and on that basis assert that all ·species of· animals have a nervous system. This looks like a generalisation, but in fact it merely affirms of •all what has already been affirmed of •each, so it tells us nothing that we didn’t already know. This ‘conclusion’ means the same as ‘All known ·species of· animals have a nervous system’. Don’t confuse this case with the following quite distinct one. Our observations of the various species of animals have revealed to us a law of animal nature, putting us in a condition to say that a nervous system will be found even in ·species of· animals that haven’t yet been discovered. This is indeed an induction, in which the conclusion is a general proposition containing more than the sum of the special propositions from which it is inferred. The difference between these two is further marked by the fact that the latter of them—the genuine induction—could be legitimate even if we hadn’t examined every single known species of animals. . . . Returning to the earlier example, think about the difference between these;

‘All the planets shine by reflected light.’

‘All planets shine by reflected light.’

The latter is an induction; the former is not. . . .

§2. Several mathematical processes should be distinguished from induction, because they are often (wrongly) called by that name, and share something important with genuine inductions, namely leading to conclusions that really are general propositions. For example, when we have proved that a straight line can’t meet a •circle at more than two points, and then successively prove the same thing of the •ellipse, the •parabola, and the •hyperbola, we can lay it down as a universal property of all conic sections. The distinction drawn in the two previous examples has no place here because there’s no difference between ‘all known conic sections’ and ‘all conic sections’, as a cone demonstrably can’t be intersected by a plane except in one of these •four lines. So we can hardly deny that the proposition arrived at is a generalisation, because there’s no room for any generalisation beyond it. But there’s no induction because there’s no inference; the conclusion is a mere summing up of the content of the four propositions from which it is drawn. The proof of a geometrical theorem by means of a diagram (on paper or in the imagination) is a bit like that though not entirely so. As I said earlier, such a demonstration doesn’t directly prove the general theorem; all it proves is that the general conclusion asserted in the theorem is true of the particular triangle or circle exhibited in the diagram. But we can see that we could prove it of any circle in the same way that we have proved it of that one; so we gather up all the singular propositions that could be thus proved, and embody them in a universal proposition. Having shown that the three angles of the triangle ABC are together equal to two right angles, we conclude that this is true of every other triangle, not •because it is true of ABC but •for the same reason that proved it to be true of ABC. The term ‘induction’ isn’t really right for this, because although its conclusion is really general it isn’t believed on the evidence of particular instances. We don’t conclude that all triangles have that property because some triangles have; rather, we accept the conclusion on the evidence that was the basis for our conviction in the particular instances.

In some mathematical arguments—so-called ‘inductions’— the conclusion does look like a generalisation based on some of the particular cases covered by it. When a mathematician has calculated a sufficient number of the terms of an algebraic or arithmetical series to have ascertained what is called the law of the series, he doesn’t hesitate to supply any number of the succeeding terms without repeating the calculations. But I take it that he does this only when it is apparent from a priori considerations (which could be exhibited in the form of demonstration) that the way each subsequent term is formed from its immediate predecessor is the same as the way each previous term was formed from its predecessor. There are instances on record of wrong results’ being reached when a series was continued without the backing of such general considerations.

Newton is said to have discovered the binomial theorem by induction, specifically calculating that

(a + b)2 = a^2 + 2ab + b^2

and that

(a + b)3 = a^3 + 3a^2b + 3ab^2 + b^3

and so on, and comparing all those results until he detected the general relation that the general binomial theorem expresses concerning the general form (a + b)n for all values of n. It’s likely enough that he did; but a mathematician like Newton, who seemed to leap to principles and conclusions that ordinary mathematicians reached only by a succession of steps, certainly couldn’t have performed the comparison in question without being led by it to the a priori ground of the law; since anyone who understands multiplication well enough to venture on multiplying several lines of symbols at one operation can’t help seeing that in raising a binomial to a power the coefficients must depend on the laws of permutation and combination; and as soon as that is recognised the theorem is demonstrated. Indeed, once it was seen that the law prevailed in a few of the lower powers, its identity with the law of permutation would at once suggest the considerations that prove it to hold universally. So even cases like this are only examples of what I have called ‘induction by parity of reasoning’, i.e. not really induction because it doesn’t involve inference of a general proposition from particular instances.

§3. It is really important to clear up a third improper use of the term ‘induction’, because •the theory of induction has been greatly confused by it, and •the confusion shows up in the most recent and elaborate treatise on the inductive philosophy that exists in our language. The error in question is that of failing to distinguish •an induction from a set of observed phenomena from •a mere description of them in general terms.

Take a phenomenon consisting of parts that can only be observed separately, as it were piecemeal. When the observations have been made, there’s a convenience (amounting for many purposes to a necessity) in getting a representation of the phenomenon as a whole by piecing these detached fragments together. A navigator sailing the ocean meets land; he can’t by any one observation determine whether it’s a continent or an island; but he coasts along it, and after a week sees that he has sailed completely round it, and then declares it to be an island. There was no particular time or place of observation at which he could see that this land was entirely surrounded by water; he learned this fact by a succession of partial observations, and then chose a general expression—‘·It’s an island·’—which summed up in three words the whole of what he observed during that week. Is there anything in the nature of an induction in this process? Did he infer something that hadn’t been observed from something that had? Certainly not. He had observed the whole of what the proposition asserts. That this land is an island isn’t an inference from the partial facts that the navigator saw in the course of his circumnavigation; it is

•the facts themselves,

•a summary of those facts,

•the description of a complex fact to which those simpler ones are as the parts of a whole.

I don’t think there is any difference in kind between this simple operation and the one by which Kepler ascertained the nature of the planetary orbits; and Kepler’s operation— or anyway all that was characteristic in it—was no more inductive than the navigator’s.

Kepler aimed to determine the real path followed by each of the planets. (Let’s take Mars, because that was the subject of the two of his three laws that didn’t require a comparison of planets.) The only way to do this was by direct observation; and all that observation could do was to ascertain many of the successive places—or rather, apparent places—of the planet. The unaided senses could establish this much:

•The planet successively occupied all these positions, or anyway positions that produced the same impressions on the eye, and

•It passed from one of these to another insensibly, with no apparent break in the continuity.

What Kepler did beyond this was to find what sort of a curve these different points would make if they were all joined together. He expressed the whole series of the observed places of Mars by the general conception of an ellipse. This operation was much harder than that of the navigator who expressed the series of his observations on successive points of the coast by the general conception of an island. But it’s the very same sort of operation; and if the navigator’s operation is not an induction but a description, this must also be true of Kepler’s.

The only real induction consisted in inferring that because the observed places of Mars were correctly represented by points in an imaginary ellipse, therefore Mars would continue to revolve in that same ellipse; and in concluding that the positions of the planet between two observations must have coincided with the intermediate points of the curve. These were inferences from the observations—facts inferred, not facts seen—·so they involved genuine induction·. But these inferences, far from being a part of what Kepler did, had been conducted long before he was born. Astronomers had long known that the planets periodically returned to the same places. With this established, there was no induction left for Kepler to make; he merely applied his new conception to the inferred facts as well as to the observed facts. When he found that an ellipse correctly represented the past path, he knew that it would represent the future path; in finding a compendious expression for the one set of facts, he found one for the other. But that’s all he found—the •expression only, not the •inference—and this didn’t add anything to the power of prediction already possessed.

§4. Whewell has given an apt name, the ‘colligation of facts’, to the descriptive operation that enables a number of details to be summed up in a single proposition. I fully agree with most of what he says about that mental process, and would gladly transfer all that part of his book into my own pages. But I think he makes one mistake, namely treating this kind of operation as the central, primary kind of induction, presenting the principles of mere colligation as principles of ‘induction’. In fact, colligation is not ‘induction’ at all in the old and accepted meaning of the word.

Whewell maintains that the general proposition that binds together the particular facts and makes them into one fact is not the mere sum of those facts but something more, because it introduce a mental conception that didn’t exist in the facts themselves. He writes:

‘The particular facts are not merely brought together, but a new element is added to them by the very act of thought by which they are combined. . . When the Greeks, after long observing the motions of the planets, saw that these motions could be considered as produced by the motion of one wheel revolving inside another wheel, these wheels were creations of their minds added to the facts they perceived by sense. Even if the wheels were no longer supposed to be material, but were reduced to mere geometrical spheres or circles, they were still products of the mind alone—something additional to the facts observed. The same is the case in all other discoveries. The facts are known, but they are insulated and unconnected until the discoverer supplies from his own store a principle [see Glossary] of connection. The pearls are there, but they won’t hang together until someone provides the string.’

In this passage Whewell indiscriminately blends together examples of both the processes that I am trying to distinguish. When the Greeks abandoned the supposition that the planetary motions were produced by the turning of material wheels, and fell back on the idea of ‘mere geometrical spheres or circles’, more was going on than the mere substitution of an ideal curve for a physical one. There was the abandonment of a •theory, and the replacement of it by a mere •description. No-one would call the doctrine of material wheels a mere description! That doctrine was an attempt to identify the force by which the planets were acted on and compelled to move in their orbits. But when the •materiality of the wheels was discarded and only their •geometrical forms retained, this was a great step in philosophy in which the attempt to account for the motions was given up and what was left of the theory was a mere description of the orbits. The proposition that

the planets were carried round by wheels revolving inside other wheels

was replaced by the proposition that

the planets moved in the lines that would be followed by bodies carried by wheels within wheels.

This was a mere way of representing the sum of the observed facts; and Kepler’s was another (and better) way of representing the same observations.

It’s true that for these merely descriptive operations, as well as for the erroneous inductive one, a mental conception was required. The conception of an ellipse had to present itself to Kepler’s mind before he could identify the planetary orbits with it. Whewell sees the conception as something added to the facts. He implies that Kepler put something into the facts by his way of conceiving them. But Kepler did no such thing. The ellipse was in the facts before Kepler recognised it; just as the island was an island before it had been sailed around. Kepler didn’t put what he had conceived into the facts, but saw it in them. A conception implies, and corresponds to, something conceived; and though the conception itself is not in the facts but in our mind, if it is to convey any knowledge about •the facts it must be a conception of something that really is in •them, some property that they actually have and that they would show to our senses if our senses were able to take cognisance of it. Suppose that Mars left behind it a visible track, and that an observer was in a fixed position that let him see the whole plane of the orbit at once; he would see it to be an ellipse. . . . I don’t think anyone would deny that in this case identifying the planet’s path with an ellipse is describing it; and I can’t see why it makes any difference that in fact that path of Mars is not directly an object of sense, given that every point in it is as exactly ascertained as if it were so.

. . . .I don’t think that the role of conceptions in the operation of studying facts has ever been overlooked or undervalued. No-one ever disputed that in order to reason about something we must have a conception of it; or that when we include a multitude of things under a general expression the expression implies a conception of something common to those things. But it by no means follows that the conception must be pre-existent, or that the mind constructs it out of its own materials. If the facts are rightly classed under the conception, that’s because the facts themselves contain something x of which the conception is a copy; and if we can’t directly perceive x, that’s because of the limited power of our organs and not because x isn’t there. The conception itself is often obtained by abstraction from the very facts which. . . .it is afterward called in to connect. Whewell admits this himself when he observes. . . .what a great service it would be to the science of physiology if a philosopher were to ‘establish a precise, tenable, and consistent conception of life’. Such a conception has to be abstracted from the phenomena of life itself—from the very facts that it is required to connect. In other cases, instead of •collecting the conception from the phenomena we are trying to colligate, we •select it from among the conceptions that have already been collected by abstraction from other facts. That’s what happened with Kepler’s laws. The facts were out of the reach of any observation that could enable the senses to identify the path of the planet directly, so the required conception couldn’t be collected by abstraction from the observations themselves; the mind had to supply hypothetically, from among the conceptions it had obtained from other portions of its experience, some one conception that would correctly represent the series of the observed facts. It had to form a supposition regarding the general course of the phenomenon, and ask itself ‘If this is the general description, what will the details be?’, and then compare these with the details actually observed. If they agreed, the hypothesis would serve as a description of the phenomenon; if not, it had to be abandoned and another tried. It’s this sort of case that gives rise to the ·false· doctrine that the mind in forming the descriptions adds something of its own that it doesn’t find in the facts.

Mars does follow an ellipse; that is a fact, surely, and one that we could see if we had adequate visual organs and a suitable position. Lacking these advantages but possessing the conception of an ellipse. . . ., Kepler looked to see whether the observed places of the planet were consistent with such a path. He found they were so; which led him to assert as a fact that the planet moves in an ellipse. But this fact, which Kepler didn’t add to the motions of the planet but found in them, was the very fact whose separate parts had been separately observed; it was the sum of the different observations.

Having stated this basic difference between my opinion and Whewell’s, I must add that his account of how a conception is selected that is suitable to express the facts seems to me absolutely right. The process is tentative: it consists of a series of guesses of which many are rejected until eventually one is found that is fit to be chosen. We know from Kepler himself that before hitting upon the ‘conception’ of an ellipse he tried nineteen other imaginary paths which he had to reject because they didn’t fit the observations. Whewell is right in saying that the successful hypothesis, though it is a guess, is usually not a lucky guess but a skillful one. The guesses that give mental unity and wholeness to a chaos of scattered particulars seldom occur except in minds abounding in knowledge and disciplined in intellectual combinations.

The tentative method is indispensable to the •colligation of facts for purposes of description. How far can it be applied to induction itself? and what functions does it have in that department? I’ll discuss this in chapter 14 of this Book. Right now my main task is to distinguish colligation from induction properly so-called; and to make that distinction clearer I’ll discuss a curious and interesting remark ·of Whewell’s· which is as strikingly true of colligation as it is false of induction, or so it seems to me.

[The ‘remark’ in question is something Whewell says about the successive accounts of the movements of (for example) the planet Mars—that it moves

•in a circle with the earth as centre;

•in a circle with the earth inside the circle but not at its centre;

•in epicycles, i.e. little circles whose centres move in a circle around the earth;

•in an ellipse.

Whewell says that each of these was correct as far as it went. In Mill’s words: ‘They all served the purpose of colligation; they all enabled the mind to represent to itself easily and all at once the whole body of facts ·about Mars· that had been established up to then.’ Mill also quotes Comte as saying the same thing, and goes on to express his own agreement and then to draw a line:] Whewell’s remark, therefore, is philosophically correct. Successive expressions for the colligation of observed facts—i.e. successive descriptions of a whole phenomenon that has been observed only in parts—can all be correct as far as they go, although they conflict with one another. But it would surely be absurd to assert this of conflicting inductions.

A scientific study of facts may be undertaken purposes of (a) the simple description of the facts, (b) the explanation of the facts, or (c) the prediction of similar facts. Of these, (a) is not, while (b) and (c) are, properly called ‘induction’. Whewell’s remark is true of (a):. . . . The elliptical theory, as a mere description, was simpler and more easily usable than its predecessors, but it wouldn’t really be more true than they were. So different (a) descriptions can all be true, but surely not different (b) explanations, such as these explanations of the movements of the planets:

(i) They are moved by a ‘virtue’ [see Glossary] inherent in their celestial nature;

(ii) They are moved by impact (which led to the hypothesis of vortices [see Glossary] as the only pushing force capable of whirling bodies in circles);

(iii) They are moved by the composition of a centripetal force with an original projectile force (Newton).

These are explanations collected by real induction from supposed parallel cases; and each had its time of being accepted. Can it be said of these. . . .that they are all true as far as they go? Isn’t it clear that at most one of them is true—that only one can be true in any degree, and the other two must be altogether false? Now consider two (c) predictions:

•Eclipses will occur when one planet or satellite casts its shadow on another;

•Eclipses will occur when some great calamity is impending over mankind.

Do these two doctrines differ only in expressing real facts with different degrees of accuracy? Assuredly one of them is true and the other absolutely false.

·START OF A LONG FOOTNOTE·

[This footnote reports and responds to two replies that Whewell made to the content of this section up to here. According to Whewell, the three explanations of planetary motion that Mill cites could all be true. His defence of this involves construing each of (i) and (ii) as being abstract and formal in such a way that (iii) can be seen as a factually contentful cashing in of it:]

‘If (i) had been maintained in such a way as to agree with the facts, the inherent virtue would have had its laws determined; and then it would have been found that the virtue related to the central body; and so, the “inherent virtue” would have coincided in its effect with (iii) the Newtonian force and the two explanations would agree—except in regard to the word “inherent”. This word indicates a part of theory (i) that was found to be untenable, so it was of course rejected in the transition to later and more exact theories.’

[Mill replies:] Whewell says that the theory of an inherent virtue agrees with Newton’s when the word ‘inherent’ is left out, which of course it would be (he says) if ‘found to be untenable’. But leave that out and where’s the theory? The word ‘inherent’ is the theory. When it is omitted, all that remains is the statement that the heavenly bodies move ‘by a virtue’—i.e. by a power of some sort—or by virtue of their celestial nature, which directly contradicts (iii) the doctrine that terrestrial bodies fall by the same law.

[Whewell again:] ‘The doctrine (ii) that the heavenly bodies were moved by vortices was successfully modified so that it came to coincide in its results with the doctrine (iii) of an inverse-quadratic centripetal force. . . When this point was reached, the vortex was merely a machinery. . . .for producing such a centripetal force, and therefore didn’t contradict the doctrine of a centripetal force. . . .’

[Mill replies:] If the doctrine (ii) of vortices had meant not that vortices existed but only that the planets moved as though they were whirled by vortices; if the hypothesis had been merely a way of representing the facts and not an attempt to account for them; if (in short) it had been only a description; no doubt it would have been reconcilable with (iii) the Newtonian theory. But the vortices were not a mere aid to conceiving the motions of the planets, but a supposed physical agent actively pushing them. . . .and according to Newton’s theory this was not true. Whewell seems to think of Newton’s theory as stating only the •directions of the forces and not •their nature, and therefore as not conflicting with any hypothesis about how they are produced. Well, (iii) the Newtonian theory regarded as a mere •description of the planetary motions doesn’t conflict with (ii); but the Newtonian theory as an •explanation of them does. The explanation consists in

ascribing those motions to a general law that holds between all particles of matter, and identifying this with the law by which bodies fall to the ground.

If the planets are kept in their orbits by a force that draws the particles composing them toward every other particle of matter in the solar system, then they are not kept in those orbits by the impulsive force of certain streams of matter that whirl them around. One explanation absolutely excludes the other. . . . Denying this is like saying that there’s no contradiction between ‘That man died because somebody killed him’ and ‘That man died a natural death’.

If Whewell is not yet satisfied, any other subject will serve equally well to test his doctrine. He will hardly say that there’s no contradiction between the members of each of these pairs:

•Light is a stream of particles

•Light is a series of waves.

•Higher organic forms arose by development from the lower.

•The different organic forms came from separate and successive acts of creation.

•Volcanoes are fed from a central fire.

•Volcanoes come from chemical action at a comparatively small depth below the earth’s surface.

If different explanations of the same fact can’t both be true, still less can different predictions. Whewell quarrels (never mind why) with my choice of example on this point, and thinks that a theory is sufficiently answered by an objection to an illustration of it. Well, examples not liable to his objection are easily found, if the production of many examples is really needed to support the thesis that conflicting predictions can’t both be true! [Mill then gives some examples.]

Whewell sees no distinction between •holding contradictory opinions on a question of fact, and •merely employing different analogies to help the conception of the same fact. Different inductions belongs to the former class, different descriptions to the latter.

·END OF LONG FOOTNOTE·

. . . .But induction is connected with colligation in two ways. •Induction is always colligation. The assertion that the planets move in ellipses was only a colligation; whereas the assertion that the planets are drawn (or tend) toward the sun was the statement of a new fact, inferred by induction. But it also served as a colligation: it brought the facts which Kepler had connected by his conception of an ellipse under the additional conception of bodies acted upon by a central force. . . . •The descriptions reached by colligation are a necessary preparation for induction. . . . Without the previous colligation of detached observations by means of one general conception we could never have obtained any basis for an induction except in the case of very limited phenomena. We couldn’t affirm any predicates of a subject that we could observe only piecemeal, let alone extending those predicates by induction to other similar subjects. . . .

§5. . . . .Whewell has replied to all this, re-stating his case but not strengthening it, as far as I can see. Since my arguments have not had the good fortune to make any impression upon him, I will add a few remarks to show more clearly what we are differing about and in some measure to account for the difference.

Nearly all writers of authority define induction as drawing inferences from known cases to unknown:

•applying to a class a predicate that has been found true of some members of the class;

•inferring from the fact that things have a certain property that other things resembling them have the same property;

•inferring from the fact that a thing had a property at a certain time that it does and will have that property at other times.

Kepler’s operation clearly wasn’t an induction in this sense of the term! The statement that Mars moves in an elliptical orbit wasn’t any kind of extension from facts to further facts. . . . Kepler didn’t extend an observed truth to cases other than those in which it had been observed; he didn’t widen the subject of the proposition expressing the observed facts. The alteration he made was in the predicate. Instead of saying ‘The successive places of Mars are so-and-so’ he summed them up in the statement ‘The successive places of Mars are points in an ellipse’. Whewell says that this statement wasn’t the sum of the observations merely, and I agree; it was the sum of the observations seen under a new point of view. But it wasn’t the sum of more than the observations, as a real induction is. It covered only cases that had been actually observed or could have been inferred from the observations before the new point of view presented itself. There was not the transition from known cases to unknown ones that constitutes ‘induction’ in the original and acknowledged meaning of the word.

Old definitions can’t prevail against new knowledge: scientific language ought to adapt itself to the true relations that hold between the things it is used to designate. If the Keplerian operation really is identical—considered as a logical process—with what happens in acknowledged induction, the definition of ‘induction’ should be widened so as to take it in. This is where I take issue with Whewell. He does think that the operations are identical. He holds that the only logical process in any induction is one that also occurs in Kepler’s case, namely guessing until a guess is found that squares with the facts. That leads him to reject all canons of induction, because it’s not by means of them that we guess. Whewell’s theory of the logic of science would be very perfect if it didn’t pass over altogether the question of proof. But I think there is such a thing as proof, and inductions relate to it quite differently from how descriptions do. Induction is proof; it is inferring something unobserved from something observed; it requires, therefore, an appropriate test of proof; and to provide that test is the special purpose of inductive logic. [That sentence is verbatim from Mill.] When on the other hand we merely collate known observations and (as Whewell puts it) connect them by means of a new conception, if the conception does serve to connect the observations we have all we want. The proposition containing it claims only to have truth that it may share with many other ways of representing the same facts. So all it requires is to be consistent with the facts; it can’t be proved and doesn’t need to be. It may serve to prove other things: it places facts into a mental connection with other facts that hadn’t previously been seen to resemble them, and thereby assimilates the case to another class of phenomena concerning which real inductions have already been made. Thus Kepler’s so-called ‘law’ brought the orbit of Mars into the class ellipse, thereby proving all the properties of an ellipse to be true of the orbit; but in this proof Kepler’s ‘law’ supplied the minor premise and not (as with real inductions) the major.

[Explaining that last clause: Mill is thinking in terms of syllogisms (dealt with in II.2) of the form:

(1) All elliptical orbits have the property F.

(2) Mars’s orbit is an elliptical orbit.

Therefore

(3) Mars’s orbit has the property F.

In this syllogisms (2)—Kepler’s ‘law’—is the minor premise (meaning that it contains the subject of the conclusion). If that syllogism expressed a real induction, Mill thinks, it would be the inductive leap from (1) to (3).]

Whewell calls something an induction if, and only if, it introduces a new mental conception; but this is running together two very different things, •invention and •proof. Introducing a new conception belongs to invention; this may be required in any operation, but it isn’t the essence of any. . . . Most inductions require no conception except what was present in each of the particular instances on which the induction is based. That all men are mortal is surely an inductive conclusion, but it doesn’t introduce any new conception; if you know that some man has died you have all the conceptions involved in the inductive generalisation. Whewell, however, considers the process of invention—i.e. forming a new conception consistent with the facts—to be not merely a necessary •part of all induction but the •whole of it.

The mental operation that extracts from a number of detached observations certain general characters in which the observed phenomena resemble one another, or resemble other known facts, is what Bacon, Locke, and most subsequent metaphysicians have understood by ‘abstraction’. I think it is strictly logically correct to call

a general expression obtained by abstraction, connecting known facts by means of shared characteristics but without inferring further facts from them

a ‘description’; and I don’t know how else anything could be described! But I don’t rely on the use of that particular word; I’m quite content to use Whewell’s term ‘colligation’, or the more general ‘mode of representing or of expressing phenomena’; provided it is clearly seen that the process is not induction but something radically different.

[Mill says that he will return to these matters, and remove difficulties that the reader may have had with the present chapter, in Book IV.]

Chapter 3. The ground of induction

§1. Thus, induction properly so-called can be briefly defined as generalisation from experience. It consists in inferring from some individual instances in which a phenomenon is observed to occur that it occurs in all instances that resemble the former in what are regarded as the material [see Glossary] circumstances.

How are we to tell material [see Glossary] circumstances from ones that are immaterial? Why are some circumstances material and others not so? I’ll come to those questions in due course. I must first point out that there’s a principle implied in the very statement of what induction is—an assumption about the course of nature and the order of the universe—namely that there are such things in nature as parallel cases; that what happens once will happen again when the circumstances are sufficiently alike, and not only again but as often as the same circumstances recur. This assumption is involved in every case of induction; and looking at the actual course of nature we see that the assumption is justified. The universe, as far as we know it, is constituted in such a way that whatever is true in any one case is true in all cases of a certain description; the only difficulty is to find what description.

This universal fact, which is our warrant for all inferences from experience, has been stated variously by different philosophers: ‘The course of nature is uniform’, ‘The universe is governed by general laws’, and the like. Metaphysicians of the school of Reid and Stewart have popularised one of the most inadequate of these formulations. The human mind’s disposition to generalise from experience—a propensity these philosophers regard as an instinct of our nature—they usually describe as something like ‘our intuitive conviction that the future will resemble the past’. Now, Bailey rightly said that time doesn’t come into it. We believe that fire will burn tomorrow because it burned today and yesterday; but we believe on precisely the same grounds that it burned before we were born and that it burns today in China. We don’t infer from the •past to the •future as such; we infer from the •known to the •unknown, from observed facts to unobserved facts. . . . This second category includes the whole region of the future; but it also includes nearly the whole of the present and of the past.

Express it how you will, the proposition that the course of nature is uniform is the basic principle or general axiom of induction. But it doesn’t explain the inductive process. On the contrary, it is itself an instance of induction, and induction that is by no means obvious. Far from being the first induction we make, it’s one of the last—or anyway one of the last that we make in a philosophically accurate form. In fact it has hardly entered into the minds of any but philosophers, and we’ll see that even they have haven’t always had a sound conception of its extent and limits. This great generalisation is, in fact, itself based on prior generalisations. The obscurer laws of nature were discovered by means of it, but the more obvious ones must have been understood and assented to as general truths before it was ever heard of. We would never have thought of saying that all phenomena conform to general laws if we hadn’t first arrived at some knowledge of many of the laws themselves—which had to be done by induction. In what sense, then, can a principle that is so far from being our earliest induction be our warrant for all the others? In the only sense in which the general propositions that we place at the head of our syllogisms ever really contribute to their validity. (I explained what this is in II.3.) Whately remarks that every induction is a syllogism with the major premise suppressed; a better formulation would be that every induction can be put into syllogistic form by supplying a major premise. If this is actually done, the principle of the uniformity of the course of nature will appear as the ultimate major premise of all inductions; so it will relate to all inductions in the way the major premise of every syllogism relates to its conclusion. And what relation is that? It doesn’t contribute to proving the conclusion; but it’s a necessary condition of its being proved, because no conclusion is proved unless there’s a true major premise.

You may want me to explain this claim that the uniformity of the course of nature is the ultimate major premise in all inductions. It certainly isn’t the immediate major premise in every inductive argument. . . . The induction, ‘John, Peter, etc. are mortal, therefore all mankind are mortal’ can be put into syllogistic form by prefixing the major premise that what is true of John, Peter, etc. is true of all mankind. How did we get this major premise? It isn’t self-evident. . . ., so we must have arrived at it by induction. [Mill says ‘by induction or ratiocination’, but he drops ratiocination without comment.] If by induction, this process like all other inductive arguments can be put into syllogistic form; and we need to construct this previous syllogism. There is in the long run only one possible construction: the real proof that what is true of John, Peter, etc. is true of all mankind can only be that a different supposition would be inconsistent with the known uniformity in the course of nature. Whether there actually would be this inconsistency may be a matter of long and delicate inquiry; but if there wouldn’t, we have no sufficient ground for the major premise of the inductive syllogism. It seems, then, that if we put the whole course of any inductive argument into a series of syllogisms, we’ll eventually arrive at an ultimate syllogism whose major premise is the principle or axiom of the uniformity of the course of nature.[1]

Why should this axiom be accepted as true? It wasn’t to be expected that thinkers would unanimously give one answer to this question, any more than with other axioms. I have already said that I think it is itself a generalisation from experience. Others think we’re compelled by the constitution of our thinking faculty to assume it as true in advance of any verification by experience. Having in II.5–6 fought at such length against a similar doctrine regarding the axioms of mathematics, using arguments that largely apply also to the present case, I’ll postpone going into more detail about it until chapter 21. At present it matters more to understand the import of the axiom itself. The proposition ‘The course of nature is uniform’ has the brevity suitable for ordinary talk rather than the precision required in philosophical language; if it’s to be accepted as true, its terms need to be explained, and given a stricter signification than they have in ordinary speech.

§2. Everyone knows that he doesn’t always expect uniformity in the course of events; he doesn’t always believe that the unknown resembles the known, that the future will resemble the past. Nobody believes that the pattern of rain and sunshine will always be the same as it is this year. Nobody expects to have the same dreams every night; indeed when the course of nature is constant in these things, everyone mentions it as something extraordinary. To look for constancy where it’s not to be expected—e.g. to expect that a date that once brought good fortune will always be a fortunate date—is rightly regarded as superstition.

The course of nature is not only uniform, it’s infinitely various. Some phenomena always recur in the combinations they had when we first met with them; others seem altogether capricious; and some get us used to experiencing them in one particular combination and then unexpectedly break that pattern. The experience of an inhabitant of Central Africa fifty years ago supported ‘All human beings are black’ as well as it supported anything. To Europeans until recently ‘All swans are white’ seemed to be an equally straightforward example of uniformity in the course of nature. Each group had to wait fifty centuries for the experience that showed them to be wrong. During that time, mankind believed in a uniformity in the course of nature where no such uniformity really existed.

According to the ancients’ notion of induction, the ‘Black person’ and ‘White swan’ conclusions involved inferences that were as legitimate as any inductions whatever. Because each conclusion was false, the ground of inference must have been insufficient, but still there was as much ground for it as this conception of induction admitted of. The induction of the ancients has been well described by Bacon under the label ‘induction by simple enumeration. . . .’. It consists in ascribing the character of general truths to all propositions that are true in every instance that we happen to know of. This is the kind of induction that is natural to a mind that isn’t used to scientific methods. The tendency (some say ‘instinct’, others say ‘association’) to infer the future from the past, the known from the unknown, is simply a habit of expecting that what has been found true once or several times, and never yet found false, will be found true again. It makes no difference whether the instances are few or many, conclusive or inconclusive; these considerations occur only on reflection. The unprompted unreflective tendency of the mind is to generalise its experience, provided it all points in one direction and no conflicting experience comes •unsought. The notion of •seeking it, •experimenting for it, •interrogating nature (Bacon’s phrase) is a much later development. When uncultivated intellects observe nature, they are passive; they accept the facts that present themselves, without actively searching for more. Only a superior mind asks itself ‘What facts do I need to come to a safe conclusion?’ and then looks out for these.

But though we always tend to generalise from unvarying experience, we aren’t always justified in doing so. We aren’t entitled to conclude that something is universally true because we have never known a counter-instance to it unless we have reason to believe that if there were any counterinstances in nature we would have known of them. When we do have this assurance—·this reason to believe. . . etc.·—this may enable induction by simple enumeration to amount practically to proof. But cases where that is how things stand are very •remarkable—I’ll discuss them in chapters 21–22 below. No such assurance can be had on any of the •ordinary subjects of scientific inquiry. Popular [see Glossary] notions are usually based on induction by simple enumeration; in science that doesn’t take us far. We’re forced to begin with it; we often have to rely on it provisionally, in the absence of anything better; but for the accurate study of nature we need a surer and a more potent instrument.

Bacon’s usual title of ‘Founder of the Inductive Philosophy’ is one he deserved primarily for pointing out the insufficiency of this rough and loose conception of induction. The value of his own contributions to a more philosophical theory of the subject has certainly been exaggerated. Although his writings contain. . . .more or less fully developed statements of several important principles of the inductive method, physical science has now far outgrown the Baconian conception of induction. Moral and political inquiry still haven’t caught up with that conception. The approved modes of reasoning on these subjects are still. . . .the very induction by simple enumeration that he condemns; and the ‘experience’ that we hear so confidently appealed to by all sects, parties, and interests is still, in Bacon’s own emphatic words, mera palpatio. [Latin for ‘mere feeling’. The kind of feeling he is referring to can be gathered from this: ‘·Those who steer by simple experience are· like men in the dark, patting the walls as they go along hoping to find their way, when they’d have done much better to wait for daylight, or light a candle, and then set off.’]

§3. For a better understanding of the problem the logician must solve if he’s to establish a scientific theory of induction, let us compare some incorrect inductions with others that are acknowledged to be legitimate. . . . That all swans are white can’t have been a good induction, because the conclusion turned out to be false. But. . . .from the earliest records, the testimony of the inhabitants of the known world was unanimous on the point. So the uniform experience of the inhabitants of the known world, agreeing in a common result with no known counter-examples, isn’t always sufficient to establish a general conclusion.

Now take an instance that is apparently not very dissimilar to this. Mankind were wrong in concluding that all swans are white; are we also wrong when we conclude that all men’s heads grow above their shoulders, and never below, in spite of the conflicting testimony of the naturalist Pliny? As there were black swans, though civilised people had existed for 3000 years on the earth without meeting with them, may there not also be ‘men whose heads do grow beneath their shoulders’, despite a rather less perfect unanimity of negative testimony from observers? [The quoted phrase is from Othello.] Most people would answer No: it’s more credible that a bird should vary in its colour than that men should vary in the relative position of their principal organs. Of course they would be right—but why are they right? We can’t answer that without going into the true theory of induction more deeply than is usually done.

. . . .When a chemist announces the existence and properties of a newly-discovered substance, if we trust his accuracy we feel assured that his conclusions will hold universally, although the induction is based on a single instance. We don’t withhold our assent until the experiment is repeated; or if we do, it’s because we aren’t sure that the one experiment was properly done; we are sure that if it was properly done it was conclusive. So here we have a general law of nature, inferred without hesitation from a single instance; a universal proposition from a singular one. Now contrast that with another case. All the instances that have been observed since the beginning of the world in support of ‘All crows are black’ wouldn’t outweigh the testimony of one apparently reliable witness who said that in a region of the earth not fully explored he had caught and examined a crow and found it to be gray.

Why is it that in some cases a single instance is sufficient for a complete induction, while in others myriads of concurring instances without a single known or presumed exception go such a very little way toward establishing a universal proposition? Whoever can answer this question knows more of the philosophy of logic than the wisest of the ancients, and has solved the problem of induction.

NOTES

[1] It needn’t be uniformity that pervades all of nature. It’s enough that it pervades the particular class of phenomena to which the induction relates. An induction about planetary motion wouldn’t be spoiled if we thought that wind and weather are the sport of chance, provided we are assuming that astronomical phenomena are governed by general laws. Otherwise the early experience of mankind would have rested on a very weak foundation, because in the infancy of science it couldn’t be known that all phenomena are regular in their course.

The major premise doesn’t have to be known in advance; it’s enough if we can now know it. . . . The conclusion ‘The Duke of Wellington is mortal’, inferred from the instances A, B, and C, implies either that we have concluded all men to be mortal or are now entitled to do so from the same evidence. A vast amount of confusion and bad logic regarding the grounds of induction would be dispelled by keeping these simple considerations in view.

Chapter 4. Laws of nature

§1. . . . .The uniformity of the course of nature is really uniformities. . . . The course of nature in general is constant because the course of each of the various phenomena that compose it is so. A certain fact invariably occurs whenever certain circumstances are present, and doesn’t occur when they are absent; the same is true of another fact; and so on. From these separate threads of connection between parts of the great whole that we call ‘nature’ a general tissue of connection unavoidably weaves itself, by which the whole is held together. If

A is always accompanied by D,

B by E and

C by F,

it follows that

A B is accompanied by D E,

A C by D F,

B C by E F, and finally

A B C by D E F.

That is how the general character of regularity is produced—a regularity which, along with and in the midst of infinite diversity, pervades all nature.

So the uniformity of the course of nature is itself a complex fact, compounded of all the separate uniformities that exist in respect of single phenomena. The •ordinary name of these various uniformities, when they are established by what we regard as a sufficient induction, is ‘laws of nature’. In •scientific usage we use ‘law of nature’ in a more restricted sense, to refer to the uniformities when reduced to their simplest expression. In the above illustration, already seven uniformities. . . .would be called ‘laws of nature’ in the everyday loose sense of that phrase; but only ·the first· three of the seven are properly distinct and independent; and when they are presupposed, the others automatically follow. So the first three are ‘laws of nature’ in the narrower sense; the other four are not, because they are mere cases of the first three, virtually included in them, and are therefore said to result from them. Anyone who affirms those three has already affirmed all the rest.

Here are three uniformities, or call them ‘laws of nature’:

(1) Air has weight,

(2) Pressure on a fluid spreads equally in all directions.

(3) Pressure in one direction, not opposed by equal pressure in the opposite direction, produces motion that doesn’t cease until equilibrium is restored.

These three uniformities should enable us to predict another uniformity, namely the rise of the mercury in the Torricellian tube [= a barometer]. This isn’t a ‘law of nature’ in the narrower sense of the phrase; it’s a result of laws of nature. It’s a case of each of the three laws; and it couldn’t be different without infringing at least one of them. If the mercury were not held up in the barometer at a height such that the column of mercury had the same weight as a column of the atmosphere of the same diameter, this would be a case of

(1) the air not pressing on the surface of the mercury with the force that is called its ‘weight’, or of

(2) the downward pressure on the mercury not being passed on equally in an upward direction, or of

(3) a body pressed in one direction and either not moving in that direction or stopping before reaching equilibrium.

So if we knew the three simple laws but had never tried the Torricellian experiment, we could have deduced its result from those laws. The known weight of the air combined with the position of the apparatus would bring the mercury within the range of the first of the three inductions; the first induction would bring it within the second, and the second within the third—all in the way I described in my account of ratiocination. We would be coming to know the more complex uniformity independently of •specific experience, through our knowledge of the simpler ones from which it results; though in due course we’ll see reasons why verification by •specific experience would still be desirable and might even be indispensable.

[Mill makes remarks about broad and narrow senses of ‘law of nature’, suggesting that the narrow sense favoured by scientists is explained by a ‘tacit reference to the original sense of “law”, namely “the expression of the will of a superior”.’ He then offers two ways of restating the narrow-sense version of the question ‘What are the laws of nature?’:]

•What are the fewest and simplest assumptions, which being granted the whole existing order of nature would result?

•What are the fewest general propositions from which all the uniformities which exist in the universe might be deductively inferred?

Every great advance that marks an epoch in the progress of science has been a step towards solving this problem. Even a simple colligation of inductions already made, with no fresh extension of the inductive inference, is already an advance in that direction. When Kepler expressed the regularity in the observed motions of the planets by the three general propositions called his ‘laws’, he was showing three simple suppositions that would suffice to construct the whole scheme of planetary motion so far as it was then known. A still greater step was made when these laws. . . .were discovered to be cases of the three laws of motion generally. . . . After this great discovery, Kepler’s three propositions, though still called ‘laws’, wouldn’t be called ‘laws of nature’ by anyone accustomed to using language with precision; that phrase would be reserved for the simpler and more general laws into which Newton is said to have resolved [see Glossary] them.

Every well-grounded inductive generalisation is either •a law of nature or •a result of laws of nature, something that could be predicted from them. And the problem of inductive logic can be summed up in two questions; how to ascertain the laws of nature; and how then to follow them into their results. But this isn’t a real analysis of the problem; it’s a mere verbal transformation of it. . . . Still, it is worth something to have reached the insight that the study of nature is the study of laws, not a law; of uniformities (plural); that the different natural phenomena have their separate rules or ways of occurring, which—though intermixed and entangled with one another—can to some extent be studied separately; that the regularity in nature is a web composed of distinct threads, and only to be understood by tracing each thread separately, for which purpose it is often necessary to unravel some portion of the web and exhibit the fibres apart. The rules of experimental inquiry are the contrivances for unraveling the web.

§2. In trying in this way to ascertain the general order of nature by ascertaining the particular order of each one of the phenomena of nature, a scientific proceeding can’t be more than an improved form of what the human understanding primitively did when not directed by science. When mankind first got the idea of studying phenomena by a method stricter and surer than the one they spontaneously started out with, they didn’t start with the supposition that nothing had yet been ascertained (Descartes’s well-meant but impracticable advice). Many of the uniformities in phenomena are so constant and so open that they force themselves on involuntary recognition. Some facts are so perpetually and familiarly accompanied by certain others that mankind learned—as children learn—to expect one where they found the other, long before they knew how to put this expectation into a proposition about there being a connection between those phenomena. No science was needed to teach that food nourishes, that water drowns or quenches thirst, that the sun gives light and heat, that bodies fall to the ground. The first scientific inquirers assumed these and their like, and set out from them to discover others that were unknown; and they weren’t wrong to do this, though they later came to see that the initial spontaneous generalisations needed to be revised when the progress of knowledge showed limits to them, or showed their truth to depend on some detail not originally attended to. You’ll see later on that there’s no logical fallacy in this procedure. Indeed, we can see that no alternative is workable, because it’s impossible to develop any •scientific method of induction, or •test of the correctness of inductions, except on the hypothesis that some inductions deserving of reliance have already been made.

For an example, look back at one of our former illustrations. With exactly the same amount of evidence (negative and positive) in each case, why did we accept the assertion that (i) there are black swans yet refuse to believe any testimony saying that (ii) some men wear their heads underneath their shoulders? Well, (i) was more credible than (ii); but why was it more credible?. . . . Apparently because there is less constancy in the colours of animals than in the general structure of their anatomy. But how do we know this? From experience, of course. It appears, then, that we need experience to tell us how much we should rely on experience, and in what cases or sorts of cases. We have to consult •experience to learn from •it when and where arguments from •it will be valid. We have no rock-bottom test to which we subject experience in general; we make experience its own test. Experience testifies that some of the uniformities it exhibits or seems to exhibit are more to be relied on than others. We have experienced uniformity U; how confident should we be that it holds in cases not yet observed? That depends on the extent to which U belongs to a class of uniformities that have hitherto been found to be uniform.

This way of correcting one generalisation by means of another—a narrower generalisation by a wider one that common sense suggests and adopts in practice—is the real type [see Glossary] of scientific induction. Skillfully contrived rules can give accuracy and precision to this process, and adapt it to all varieties of cases, but they can’t make any essential difference to its principle.

To apply a test of this sort we must already have some general knowledge of the typical character of the uniformities existing throughout nature. So the indispensable foundation of a scientific formula of induction has to be a survey of the inductions mankind has been unscientifically led to, with the special purpose of ascertaining what kinds of uniformities have been found to be perfectly invariable, pervading all nature, and what kinds have been found to vary with difference of time, place, or other changeable details.

§3. The need for such a survey is confirmed by the fact that stronger inductions are the touchstone to which we always try to bring the weaker. If we find a way to deduce a weaker induction from stronger ones, it instantly acquires all their strength; and it even adds to that strength, because the experience on which the weaker induction previously rested becomes additional evidence for the truth of the stronger ones. Suppose we infer from historical evidence that the uncontrolled power of a monarch, of an aristocracy, or of the majority, will often be abused; we can rely on this generalisation with much greater confidence when it is shown to follow from facts that are even better established—•the low degree of elevation of character ever yet attained by the average of mankind, and •the poor success-rate of most known procedures for making reason and conscience predominate over the selfish propensities. And obviously these more general facts get more evidence from what history tells us about the effects of despotism. The strong induction becomes still stronger when a weaker one has been bound up with it.

On the other hand, if an induction conflicts with stronger inductions or with conclusions that follow from them, then the weaker one must give way—unless reconsideration shows that some of the stronger inductions have been expressed with greater universality than their evidence justifies. The age-old opinion that a comet. . . .was the precursor of calamities. . . .; the belief in the truthfulness of the oracles of Delphi or Dodona; the reliance on astrology or on the weather-prophecies in almanacs; these were doubtless inductions supposed to be based on experience; and it seems that faith in such delusions can hold out against many failures, as long as it’s nourished by a reasonable number of casual coincidences between the prediction and the event.[1] What has really put an end to these insufficient inductions is their inconsistency with the stronger inductions subsequently obtained through scientific inquiry, concerning the causes that terrestrial events really depend on. In places where those scientific truths haven’t yet penetrated, the same or similar delusions still prevail.

Here is a ·two-part· general principle about any two inductions, whether strong or weak:

(1) If they can be connected by ratiocination, they tend to confirm one another.

(2) If they lead deductively to consequences that are incompatible, they become tests of each other, showing that one or other must be given up or at least limited in some way.

In case (1) the induction that becomes a conclusion from ratiocination becomes at least as certain as the weakest of those from which it is deduced; while in general all are more or less increased in certainty. Thus the mercury-in-tube experiment, though it’s a mere case of three more general laws, doesn’t just strengthen greatly the evidence on which those laws rested but raises the level of one of them (the weight of the atmosphere) from ‘doubtful’ to ‘completely established’.

Thus, if among the uniformities that have been found to exist in nature there are •some that may be considered quite certain and quite universal (so far as any human purpose requires certainty), then we may be able to use •these to raise multitudes of other inductions to the same point on the ·certainty· scale. For if we can show that either •inductive inference I_2 is true or •the certain and universal induction I_1 has an exception, then I_2 will attain the same certainty and security as I_1. . . . It will be proved to be a law; and if it’s not a result of other and simpler laws, it will be a law of nature.

There are such certain and universal inductions; and it’s because there are that a logic of induction is possible.

NOTES

[1] Whewell won’t allow these and their like to count as ‘inductions’, because such superstitious fancies ‘were not •collected from the facts by seeking a law of their occurrence, but were •suggested by an imagining of the anger of superior powers. . . .’ But the question is not ‘How were these notions first suggested?’ but ‘What evidence is. . . .supposed to support them?’ If the believers in these erroneous opinions had been challenged to defend them, they would have referred to experience: to the comet that preceded the death of Julius Caesar, or to oracles and other prophecies known to have been fulfilled. Analogous superstitions exist even today, and their hold on the believers’ minds depends on the supposed evidence of experience—·mostly consisting of casual coincidences· [see Glossary]. I admit that the influence of such coincidences wouldn’t be what it is if it weren’t strengthened by an antecedent presumption; but this is not a special feature of superstitions; preconceived notions of probability help to explain many other cases of belief on insufficient evidence. The a priori prejudice improperly predisposes the believer’s mind to interpret his experience in that way, but the believer still sincerely regards his belief as a legitimate conclusion from experience.

—My theme could easily be illustrated by cases where antecedent prejudice has no role. Whately writes: ‘For many ages all farmers and gardeners were firmly convinced •that the crops would never turn out good unless the seed was sown when the moon was waxing, and •that they had learned this from experience.’ This was induction, but bad induction; just as an invalid syllogism is reasoning, but bad reasoning.

Chapter 5. The law of universal causation

§1. The phenomena of nature exist in two relations to one another—simultaneity and succession. Every phenomenon is related in a uniform manner to •some phenomena that coexist with it and to •some that have preceded and will follow it.

Of the uniformities that exist among synchronous [see Glossary] phenomena, the most important in every way are the laws of number, closely followed by the laws of space, i.e. of extension and figure. The laws of number are common to synchronous and successive phenomena. That two and two make four is equally true whether the second two follow the first two or accompany them; it’s as true of days and years as of feet and inches. In contrast with that, the laws of extension and figure (i.e. the theorems of geometry) are laws only of simultaneous phenomena. The various parts of space and of the objects that are said to ‘fill’ space coexist, and the unvarying laws that are the subject of the science of geometry express how they coexist. [Mill is here taking ‘x coexists with y’ to mean ‘x exists at some time when y also exists’.]

To understand and prove these laws—i.e. these uniformities—you don’t have to suppose any lapse of time, any variety of facts or events succeeding one another. The propositions of geometry are independent of the succession of events. All things that have extension, i.e. that fill space, are subject to geometrical laws. Having extension they must have figure; so they must •have some figure in particular and •have all the properties that geometry assigns to that figure. An example: If x is a sphere and y a cylinder with the same height and diameter, x’s volume will be exactly two-thirds of y’s, no matter what stuff x and y are made of. Another example: Each body and each point within a body must occupy some place or position among other bodies; and the position of two bodies relatively to each other, whatever stuff they are made of, can be unerringly inferred from the position of each of them relatively to any third body.

In the laws of number, then, and in the laws of space, we clear cases of the rigorous universality that we’re looking for. Those laws have always been the type [see Glossary] of certainty, the standard of comparison for all lower degrees of evidentness. They are so perfectly invariable that we can’t even conceive any exception to them (and philosophers have been led—though wrongly—to think that what makes them evident is not experience but the basic constitution of the intellect). So if we could deduce from the laws of space and number any other kind of uniformities, that would be proof positive that those other uniformities had the same rigorous certainty. But we can’t do this. From laws of space and number alone nothing can be deduced but laws of space and number.

For us the most valuable truths about phenomena are the ones concerning the order of their succession. Our knowledge of these truths is the basis for every reasonable anticipation of future facts, and for any power we have to influence those facts to our advantage. Even the laws of geometry—·which don’t involve succession·—are chiefly of practical importance to us because they enter into premises from which we can infer the order of the succession of phenomena. The motion of bodies, the action of forces, and the propagation of influences of all sorts take place along certain lines and over definite distances; so the properties of those lines and distances are an important part of the laws to which those phenomena are subject. Similarly, •motions, •forces or other influences, and •times are numerable quantities; so the properties of number are applicable to them as to everything else. But the laws of number and space can’t unaided contribute to the discovery of uniformities of succession. They can be made to do that work only when we combine with them premises about uniformities of succession that we already know. Take for example the propositions:

•Bodies acted on by an instantaneous force move with uniform velocity in straight lines.

•Bodies acted on by a continuous force move with accelerated velocity in straight lines.

•Bodies acted on by two forces in different directions move in the diagonal of a parallelogram, whose sides represent the direction and quantity of those forces.

If we combine these truths with certain geometrical propositions (e.g. that a triangle is half a parallelogram of the same base and altitude), we can deduce another important uniformity of succession:

•A body moving around a centre of force marks off areas proportional to the times.

But we must have laws of succession in our premises if we are to reach truths of succession in our conclusions. . . .

The laws of space are only laws of simultaneous phenomena; and the laws of number, though true of successive phenomena, don’t relate to their succession; so the rigorous •certainty and •universality of these laws don’t carry through to laws of succession. We must try, then, to find some law of succession that has •those attributes, making it a fit basis for processes of discovering and testing all other uniformities of succession. This basic law must resemble the truths of geometry in their most remarkable special feature, namely that they are never ever defeated or suspended.

Of the uniformities in the succession of phenomena that common observation brings to light very •few have even an apparent claim to this rigorous indefeasibility; and of those few only •one has been completely sustained in this claim. That one, however, is a law that is universal also in another sense; it is coextensive [see Glossary] with the entire field of successive phenomena, all instances of succession being examples of it. This law is the Law of Causation. The truth that every fact that has a beginning has a cause is coextensive with human experience.

You may think that this doesn’t amount to much, because it only says ‘It’s a law that every event depends on some law’ or ‘It’s a law that there’s a law for everything’. But we shouldn’t conclude that the principle’s generality is merely verbal; when we look into it we’ll find that far from being vague or meaningless it is a most important and really fundamental truth.

§2. The notion of cause is the root of the whole theory of induction; so we must at the outset of our inquiry get it fixed with as much precision as we can manage. There is an old and still-running battle among different schools of metaphysicians concerning the •origin and •analysis of our idea of causation; but—fortunately!—we don’t need to settle that before starting our search for the true theory of induction. The science of the investigation of truth by means of evidence is •independent of many of the controversies that perplex the science of the ultimate constitution of the human mind, and •has no need to push the analysis of mental phenomenon to the extreme limit that a metaphysician ought to demand.

Thus, when in the course of this inquiry I speak of the ‘cause’ of any phenomenon, I don’t mean a cause that isn’t itself a phenomenon; I am not inquiring into the ultimate or ontological cause of anything. To adopt a distinction familiar in the writings of Reid and other Scottish metaphysicians, the causes I’m concerned with are not efficient causes [see Glossary] but physical causes. They are ‘causes’ purely in the sense in which one physical fact is said to be the ‘cause’ of another. I’m not called upon to give an opinion about the ‘efficient causes’ of phenomena, or whether there are any. According to the schools of metaphysics that are most currently most fashionable,

The notion of causation implies a mysterious and most powerful tie of a kind that can’t (or anyway doesn’t) exist between two physical facts x and y such that x is always followed by y and is popularly caused y’s ‘cause’. So if we want to find the true cause, the cause that isn’t only •followed by the effect but actually •produces it, we have to ascend higher [Mill’s phrase; we might prefer to say ‘dig deeper’] into the essences and inherent constitution of things.

I have no need to do that for the purposes of the present inquiry, and no such doctrine will be found in the following pages. The only notion of cause that the theory of induction needs is one that can be gained from experience. The Law of Causation, the recognition of which is the main pillar of inductive science, is merely the familiar truth that invariability of succession is empirically found to obtain between every fact in nature and some other fact that has preceded it—independently of any question about •the ultimate ·or absolutely basic· cause of phenomena or about •the nature of ‘things in themselves’.

So there’s an invariable order of succession between phenomena existing at any instant and the phenomena that exist at the next instant. . . . Certain facts always are—and, we believe, always will be—followed by certain other facts. The invariable antecedent is termed the ‘cause’, and the invariable consequent the ‘effect’. And the law of causation holds universally because every consequent is connected in this way with some particular antecedent or set of antecedents. Every fact that has begun to exist was preceded by some fact(s) with which it is invariably connected. For every event E there’s some combination of objects or events—some combination of circumstances, positive and negative—the occurrence of which is always followed by E. Even when we don’t yet know what this combination is for a given E, we never doubt that there is one, and that it never occurs without E as its effect or consequence. If this truth weren’t universal, we couldn’t express the inductive process in rules. . . .

§3. This invariable sequence seldom if ever holds between a consequent and •a single antecedent. It’s usually between a consequent and •the sum of several antecedents, the concurrence [see Glossary] of all of them being needed to produce—i.e. to be certain of being followed by—the consequent. People often single out one of the antecedents as the ‘cause’ and call the others merely ‘conditions’. [Mill elaborates this in more detail than we need. Someone dies because of his eating some oysters. Many will say that his eating the oysters was ‘the cause’ of his death, but other things were also needed: his general physical constitution, his state of health at this moment, perhaps the room-temperature, etc. These plus the eating of the oysters made up the cause of his death: •the other causes were waiting to have the oyster-meal added to them so that the effect could be produced. •They tend to be left out of ‘the cause’ because they were relatively long-lasting states of affairs and not short-term events like the eating of the meal. Because the total cause was topped up by that one event, people get the impression that the event had ‘a more immediate and close connection’ with the death than did the other conditions; but it didn’t.]

Even when we’re aiming at accuracy we don’t list all the conditions, but that’s because some of them are understood without being expressed, or because our immediate purpose won’t be harmed by omitting them. When we say that the cause of a man’s death was that •his foot slipped when he was climbing a ladder, we omit •his weight as part of the clause (though it related to his death in the same way as his foot-slip did) because there was no need to mention it in this context. . . . When the decision of a legislative assembly is settled by the casting vote of the chairman, we sometimes say that he was the cause of everything that resulted from the enactment. We don’t really think that his single vote contributed more to the result than any other affirmative vote; but for our present purpose, namely to insist on his individual responsibility, the part that anyone else had in the transaction is not material.

In all these examples, the fact that was picked out as ‘the cause’ was the condition that came into existence last. But don’t think that in the use of ‘the cause’ we are strictly guided by this or any other rule. There’s no scientific basis for the distinction between the •cause of a phenomenon and its •conditions; you can see this from how capriciously we select the condition of an event that we choose to call its ‘cause’. However many conditions there are, almost any of them might count as ‘the cause’ because our immediate purpose can afford to neglect the others. Take a case where a stone thrown into water sinks to the bottom. What are the conditions of this event? In the first place there must be •a stone, and •water, and •the stone’s being thrown into the water; but these suppositions are part of the statement of what the event is, and it’s bad, a tautology, to include them among the ‘conditions’ of the event. This class of conditions have never been called ‘cause’ except by the Aristotelians, who called them ‘the material cause’. [Actually, they’d have said this about the stone and the water, but not about the stone’s being thrown into the water.] The next condition is there being •an earth; and accordingly it’s often said that the fall of a stone is caused by the earth, i.e. by

•a power of the earth,

•a property of the earth, or

•a force exerted by the earth,

all of which are merely roundabout ways of saying that it is caused by the earth. Or the fall may be said to be caused by

•the earth’s attraction.

That’s a technical way of saying that the earth causes the motion, with an extra special feature, namely that the motion is toward the earth—which is a feature of the effect, not of the cause. Now pass to another condition: it’s not enough that the earth should exist; the body must be close enough to the earth for the earth’s attraction to outweigh the attraction of any other body. So we can correctly say that the cause of the stone’s falling is its being within the sphere of the earth’s attraction. A further condition: because the stone is immersed in water, it can’t reach the bottom unless its specific gravity exceeds that of the water, i.e. unless it weighs more than an equal volume of water. So it would be regarded as correct to say that the cause of the stone’s going to the bottom was its exceeding in specific gravity the fluid in which it was immersed.

Thus we see that each condition of the phenomenon may be taken in its turn and spoken of as if it were the entire cause—with equal propriety in everyday speech and equal impropriety in scientific discourse. The particular condition picked out as ‘the cause’ is usually the one •whose share in the event is superficially the most conspicuous, or •whose status as required for the event we happen to be emphasising at the moment. This second consideration can even lead us to select as ‘the cause’ one of the negative conditions, as in ‘ The army was taken by surprise because the sentinel was off his post’. The sentinel’s absence didn’t create the enemy or put the soldiers to sleep, so how did it cause them to be surprised? All that is really meant is that the event wouldn’t have happened if he had been at his duty. His being off his post was not

•a producing cause, but merely

•the absence of a preventing cause.

It was simply equivalent to his non-existence. No consequences can come from nothing, from a mere negation. All effects are connected by the law of causation with some set of positive conditions, though negative ones are almost always required in addition. In short: every fact or phenomenon that has a beginning arises when some certain combination of positive facts exists, provided certain other positive facts do not exist.

We tend to associate the idea of the cause of E_2 with the event E_1 that immediately precedes E_2, rather than with any of the earlier states—i.e. permanent facts—that are also conditions of E_2. (You can see this in the example of death caused by eating oysters.) The reason for this tendency is that E_1 begins to exist immediately before E_2, whereas the other conditions may have pre-existed for an indefinite time. We see this tendency at work in the different logical fictions that even men of science resort to so as not to give the name ‘cause’ to anything that existed for an indeterminate length of time before the effect. Thus, rather than saying that the earth causes the fall of bodies, they ascribe it to a force exerted by the earth or an attraction by the earth; and they think of these abstractions as used up by each effort and therefore constituting at each successive instant a new fact that is simultaneous with the effect or immediately preceding it. . . .

·START OF A LONG FOOTNOTE·

An intelligent reviewer of this work in the Prospective Review [R. H. Hutton] disputes my thesis that any condition of a phenomenon may be—and on some occasions and for some purposes actually is—spoken of as ‘the cause’. He says: ‘We always apply the word “cause” to the element in the antecedents that exercises force. . . .’ Also: ‘everyone would feel’ it to be wrong to say that the cause of a surprise was the sentinel’s being off his post, but would feel that the ‘allurement or force which drew him off his post might be so called. . . .’. I can’t think that of these two:

•The event occurred because the sentinel was absent

•The event occurred because the sentinel was bribed to be absent

one is wrong and the other right. The only direct effect of the bribe was his absence, so the bribe could be called the remote cause of the surprise; but only because the absence was the immediate cause. I don’t think anyone would accept one expression and reject the other unless he had a theory to support.

[The reviewer claimed that several statements implied by Mill’s account—e.g. that a man’s having bodily organs was part of the cause of his dying when he took poison—are things that no-one would say. Mill accepts this, and patiently repeats his explanation of why such things sound wrong though they are true. He continues:]

As for the assertion that nothing is called ‘the cause’ unless it exerts active force: I’ll set aside the question of what ‘active force’ means, and will use the phrase in its popular sense. Well, then, of these two—

•He fell because his foot slipped in climbing a ladder

•He fell because of his weight

—which sounds better? The active force bringing about his fall was his weight, not the motion of his foot! [Mill gives other examples in which the most intuitively-plausible candidate for the role of ‘the cause’ is not the force-exerting one, ending with:] The opening of flood-gates is said to be the cause of the flow of water; yet the active force is exerted by the water itself, and opening the flood-gates merely supplies a negative condition. The reviewer adds: ‘Relations of space and time are absolutely passive conditions yet are absolutely necessary to physical phenomena; but no-one ever applies the word “cause” to these without being immediately stopped by those who hear him.’ I have to disagree even with this. Few people would feel that it was wrong or strange to say that a secret became known because it was spoken of when X was within earshot (a condition of space), or that the cause why this tree is taller than that one is that it has been longer planted (a condition of time).

·END OF THE LONG FOOTNOTE·

Philosophically speaking, then, the cause is the sum total of the conditions, positive and negative—the whole of the contingencies of every sort from which the consequent invariably follows. But the negative conditions of any phenomenon can be all summed up in one phrase, ‘the absence of preventing or counteracting causes’, which spares us the wordy labour of listing them separately. The convenience of this form of expression is mainly based on the fact that in most cases cause C_1’s effects in counteracting cause C_2 can with strict scientific exactness be regarded as a mere extension of C_1’s own proper and separate effects. When gravity retards the upward motion of a projectile and deflects it into a different trajectory, it is producing the very same •kind of effect—and even (as mathematicians know) the same •quantity of effect, as it does when causing an unsupported body to fall to the ground. When an alkaline solution mixed with an acid destroys its sourness, and prevents it from reddening vegetable blues, it’s because the specific effect of the alkali is to combine with the acid and form a compound with totally different qualities. Causes of all sorts have this property of preventing the effects of other causes by virtue of the same laws according to which they produce their own.[1] This enables us to do without any mention of negative conditions: we can •establish the general axiom that all causes are liable to be counteracted in their effects by one another, and •limit the notion of cause to •the sum of the positive conditions, and

•the sum of the positive conditions, and

•one negative condition, always the same one, namely the absence of counteracting causes;

·and just because the negative condition is always the same it can be silently understood, and in that spirit dropped from the story·.

§4. . . . .In most cases of causation a distinction is commonly drawn between •something that acts and •some other thing that is acted upon; between an •agent and a •patient [see Glossary]. Everyone would agree that both of these are •conditions of the phenomenon, but it would be thought absurd to call the latter the ‘cause’, that label being reserved for the former. But when we look into this we find that this distinction vanishes, or rather turns out to be only verbal. Its source is a mere fact about wording: the object that is said to be acted on—and is regarded as the scene in which the effect occurs—is usually included in the phrase by which the effect is spoken of, so that if it were also counted as part of the cause there would be the appearance of something’s being incongruously said to cause itself. Return to falling bodies, and the question: ‘What is the cause that makes a stone fall?’ If the answer had been ‘the stone itself’, that would seem to contradict the meaning of the word ‘cause’. So the stone is conceived as the patient, and the earth is represented as the agent or cause. [Mill wrote ‘the earth (or, according to the common and most unphilosophical practice, an occult [see Glossary] quality of the earth)’.] But this is a superficial matter: we can conceive the stone as causing its own fall, as long as we word this so as to avoid the mere verbal incongruity. We might say: ‘The stone moves toward the earth by the properties of the matter composing it’; and then there’s nothing wrong with calling the stone itself the ‘agent’. (Wanting to save the established doctrine that matter is inactive, men have usually preferred to say that the cause is not •the stone itself but •its weight or gravitation—an occult quality.)

Those who have defended a radical distinction between agent and patient have generally had this thought:

The ‘agent’ is what causes some state of, or some change in the state of, another object that is called the ‘patient’.

But a little reflection will show that our way of speaking of phenomena as states of the various objects that take part in them. . . .is simply a sort of logical fiction, sometimes useful as one among several formulations, but never to be mistaken for an expression of a scientific truth. Even the attributes of an object that might seem with greatest propriety to be called ‘states’ of the object—its sensible qualities, its colour, hardness, shape, and the like—are really. . . .phenomena of causation, in which the substance is the agent or producing cause, and the patient is our own organs and those of other sentient beings. What we call ‘states of’ objects are sequences ·of events· into which the objects enter, usually as antecedents or causes; and things are never more active than when they are producing the phenomena in which they are said to be ‘acted on’! According to the theory of gravitation, a falling stone is as much an agent as the earth is—the earth attracts the stone but is also attracted by it. When a sensation is produced in our organs, the laws of our organisation—and even the laws of our minds—are as directly operative in determining the effect as are the laws of the external object. We call prussic acid the ‘agent’ of a person’s death, but the whole of his vital and organic properties are as actively instrumental as the poison in the chain of effects that kills him. In the process of education, we may call the teacher the ‘agent’, and the pupil the material that is acted on; but actually all the facts that pre-existed in the pupil’s mind act either for or against the teacher’s efforts. The agent in vision isn’t light alone but light coupled with the active properties of the eye and brain, and with those of the visible object. The ‘agent’/‘patient’ distinction is purely verbal; patients are always agents; in most natural phenomena, indeed, they are agents to such a degree that they react forcibly on the causes that acted on them. . . . All the positive conditions of a phenomenon are alike agents, alike active; and in any account of the cause that professes to be complete, none of them should be excluded except ones that have already been implied in the words used for describing the effect. . . .

§5. I should deal separately with the case of causation where the effect is to invest an object with a certain property—i.e. not to produce a certain phenomenon but to fit something else for producing it. When sulphur, charcoal, and nitre are put together in a certain way and in certain proportions, the effect is not an explosion but a mixture that is explosive—i.e. that will explode in certain circumstances. The various natural and artificial causes that educate the human body or the human mind have for their principal effect not •to make the body or mind immediately do anything but •to endow it with certain properties—i.e. to ensure that in certain circumstances certain results will take place •in it or •as consequences of it. . . . Painting a wall white doesn’t merely produce the sensation of white in those who see the wall; it confers on the wall the permanent property of giving that kind of sensation. In relation to the sensation, the painting of the wall is a condition of a condition; it is a condition of the wall’s causing that particular fact. The wall may have been painted years ago, but it has acquired a property that has lasted till now, and will last longer; the antecedent condition needed to enable the wall to become in its turn a condition has been fulfilled once for all. In a case like this, where the immediate effect is a property produced in an object, no-one these days thinks that the property is a substantive entity—·a special kind of thing·—‘inherent’ in the object. What has been produced could be called a state of preparation in an object for producing an effect. . . . In the case of the gunpowder •this state of preparation consists in the particles’ coming to be close to one another. In the example of the wall, •it consists in a new spatial closeness of the wall to the paint. In the example of the moulding influences on the human mind, •its involving spatial relations is only conjectural; even if we assume the materialistic hypothesis, there’s still a question as to whether the increased ease with which the well-trained brain sums up a column of figures is a result of some permanent new arrangement of some of its material particles. So we must content ourselves with what we know, and include among the effects of causes the capacities given to objects of being causes of other effects. A ‘capacity’ isn’t a real thing existing in the object; it’s merely a name for our belief that the object will act in a certain way if certain new circumstances arise. We can give this assurance of future events a fictitious objective existence by calling it ‘a state of the object’. But unless the state consists in a spatial arrangement of particles (as with the gunpowder), it expresses no present fact and is merely a contingent future fact re-presented under another name. . . .

§6. I now present a distinction that is of first-rate importance both for clarifying the notion of cause and for blocking a plausible objection that is often brought against the view I have taken of the subject.

When I define the cause of x—in the only sense in which this book has any concern with causes—to be ‘the antecedent that x invariably follows’ I do not mean ‘the antecedent that x invariably has followed in our past experience’. An account of causation in terms of ‘has followed’ would be open to Reid’s very plausible objection that then night must be the cause of day, and day the cause of night, because day and night have invariably followed one another from the beginning of the world. It’s essential to our use of ‘cause’ that we should believe not only •that the antecedent always has been followed by the consequent, but that as long as the present constitution of things[2] endures, it always will be so. And this isn’t true of day and night. We believe that night will be followed by day not

•under all imaginable circumstances, but only

•provided the sun rises above the horizon.

If the sun stopped rising (and for all we know, its doing so may be perfectly compatible with the general laws of matter), night could be eternal. On the other hand, if the sun is above the horizon, its light not extinct, and no opaque body between it and us, we firmly believe that unless there’s a change in the properties of matter •this combination of antecedents will always be followed by day; that •if this combination were indefinitely prolonged, it would always be day; and that •if the same combination had always existed, it would always have been day, quite independently of night as a previous condition. That’s why we don’t call night the ‘cause’ of day, or even a ‘condition’ of day. The only conditions of day are •the existence of the sun (or some such luminous body), and •there being no light-blocker in a straight line[3] between that body and the part of the earth where we are situated; and the combination of these, without any superfluous details, constitutes the cause. This is what writers mean when they say that the notion of cause involves the idea of necessity. If ‘necessity’ has any agreed meaning it is unconditionalness. To say that x is necessary, that x must be, is to say that x will be, no matter what else happens. The succession of day and night is obviously not ‘necessary’ in this sense, because it is conditional on the occurrence of other antecedents. If x will be followed by y when and only when z is the case, x isn’t the cause of y even if no instance of x has ever occurred without y following it.

. . . .So we can define the cause of a phenomenon to be the antecedent (or combination of antecedents) which it invariably and unconditionally follows. Or if we adopt the convenient usage in which ‘cause’ is confined to the combination of positive conditions, then we must replace ‘unconditionally’ by ‘subject to no conditions except negative ones’.

Some may want to object:

‘The sequence of night and day is invariable in our experience; we have as much ground in •this case as experience can give in •any case for recognising the two phenomena as cause and effect. To say that more is necessary—to require a belief that the succession is unconditional—is to admit that causation involves an element of belief that isn’t derived from experience.

I answer that it is experience itself that teaches us that one uniformity of sequence is conditional and another unconditional. When we judge that the succession of night and day is a derivative sequence, depending on something else, we are going by experience. It’s the evidence of experience that convinces us that day could exist without being followed by night, and night without being followed by day. To say as Tulloch does that these beliefs are ‘not generated by our mere observation of sequence’ is to forget that twice in every 24 hours, when the sky is clear, we have decisive evidence that the cause of day is the sun. We have empirical knowledge of the sun that justifies us on empirical grounds in concluding that if the sun were always above the horizon there would ·always· be day even if there had been no night, and that if the sun were always below the horizon there would ·always· be night even if there had been no day. [Mill adds a reminder that if x is only a conditionally invariable antecedent of y, then x’s status as an invariable antecedent of y is fragile.]

[Mill now offers a paragraph explaining how a combination of causes (unconditionally invariable antecedents) can generate conditionally invariable relations. He uses this to rebut something said by a contemporary philosopher. And other contemporaries—especially Whewell—are criticised in a further paragraph that is all about terminology.]

§7. Does a cause always relate to its effect as antecedent to consequent? Don’t we often speak of two simultaneous facts as cause and effect—fire as the cause of warmth, the sun and moisture as the cause of vegetation, and the like? Since a cause doesn’t necessarily go out of existence because its effect has been produced, the two do very generally coexist; and there are appearances, and common expressions, that seem to imply not only that causes can but that they must be contemporaneous with their effects. The scholastics have had as a dogma ‘When the cause ceases, so does the effect’ [Mill gives this in Latin], and there was a time, it seems, when it was generally believed that the continuance of an effect requires the continued existence of the cause. Kepler’s numerous attempts to explain the motions of the planets on mechanical principles were doomed by his always supposing that the agency that set them in motion must continue to operate in order to keep up the motion it at first produced. Yet there have always been many familiar examples of effects continuing long after their causes had ceased. Sun-stroke gives a person brain-fever; will the fever go off as soon as he is moved into the shade? A sword is run through someone’s body; must the sword remain in his body for him to continue to be dead? Once a plough has been made, it remains a ploughshare without any continuance of heating and hammering. . . . On the other hand, the pressure that forces up the mercury in a vacuum-tube must be continued in order to keep it up there. This (it may be replied) is because another force—the force of gravity—is acting continually, and would bring the mercury down again if it weren’t counterbalanced by an equally constant force. Well, then: a tight bandage causes pain, which will sometimes stop as soon as the bandage is removed. The illumination that the sun diffuses over the earth ceases when the sun goes down.

So there’s a distinction to be drawn. Sometimes the conditions needed for the first production of a phenomenon are also needed for its continuance; though more often its continuance requires no condition except negative ones. •Most things, once produced, continue as they are until something changes or destroys them; but •some require the permanent presence of the agencies that produced them at first. We could choose to regard these as instantaneous phenomena, needing to be renewed at each instant by the cause that first generated them. The illumination of any given point of space has always been regarded as an instantaneous fact, which perishes and is perpetually renewed as long as the required conditions obtain. This way of talking spares us the necessity of admitting that the continuance of a cause is ever required to maintain the effect; because we can say that the cause is required not to •maintain the effect but to •reproduce it or else to •counteract some force tending to destroy it. This may be a convenient terminology, but that’s all it is—terminology. The fact remains that in some cases (though only a minority) the continuance of the conditions that produced an effect is necessary for the continuance of the effect.

Is it strictly necessary that a cause should precede—by ever so short an instant—the production of its effect? For my present purposes this doesn’t matter. There certainly are cases where the effect follows with no interval that we can detect; and when there is an interval we can’t tell how many intermediate links, imperceptible to us, may fill it up. But even if an effect can start simultaneously with its cause, this doesn’t affect the view of causation that I am defending. Whether or not the cause and its effect must be successive, •it’s the beginning of a phenomenon that implies a cause, and •causation is the law of the succession of phenomena. If these ·two· axioms are granted, we can drop the words ‘antecedent’ and ‘consequent’ as applied to cause and effect, though I don’t see any need to do so. I have no objection to defining a cause as an assemblage of phenomena such that, when it occurs, some other phenomenon invariably starts or has its origin. It doesn’t matter whether the effect •coincides with the last of its conditions, or •immediately follows it. It doesn’t precede it; and when we are wondering which of two coexistent phenomena is the cause and which the effect, we rightly regard the question as answered if we can ascertain which of them came first.

§8. . . . .A single phenomenon is often seen to be followed by several different sorts of effects that happen simultaneously. . . .

•The sun produces the motions of the planets, daylight, and heat.

•The earth causes the fall of heavy bodies, and the phenomena of the magnetic needle.

•A crystal of galena [lead sulphide] causes the sensations of hardness, weight, cubic shape, gray colour, and many others between which we can trace no interdependence.

The terminology of ‘properties’ and ‘powers’ is specially adapted to this sort of case. When a single phenomenon is followed. . . .by effects of radically different kinds, it’s usual to say that each different sort of effect is produced by a different ‘property’ of the cause. Thus we distinguish the gravitational ‘property’ of the earth from its magnetic ‘property’; the gravitational, light-making and heat-making ‘properties’ of the sun; a crystal’s ‘properties’ of colour, shape, weight, and hardness. These are mere phrases: they don’t explain anything or add anything to our knowledge of the subject; but considered as abstract names denoting the connection between an object and the different effects it produces, they’re a powerful instrument of abridgment and of the acceleration of thought that abridgment brings about.

All this leads to a conception that we’ll find to be important, namely that of a permanent cause, or original [see Glossary] natural agent. A number of permanent causes have existed for as long as the human race has, and indeed longer—probably vastly longer. The sun, the earth, and the planets are such permanent causes, as are their various constituents—air, water, and other simple or compound substances of which nature is made up. These have existed, and their effects have taken place, from the very beginning of our experience. But we can’t explain the origin of the permanent causes themselves. Why these particular natural agents and no others existed originally, why they occur in such-and-such proportions, why they are distributed in such-and-such a way throughout space—we don’t know the answers to any of this. Furthermore, we can’t discover anything regular in the distribution itself. Given how these causes or agents are distributed in •one part of space, there is no uniformity or law that will let us conjecture what the distribution is •elsewhere. So the coexistence of this and that primeval cause is a mere casual coincidence, so far as we are concerned; so we don’t classify as a case of causation or a law of nature any regularity (of following or coexisting) between the effects of this one and the effects of that one. We have no basis for expecting such regularities, except where we have direct evidence about how the relevant natural agents—the things on whose properties the regularities ultimately depend—are distributed in space. These permanent causes aren’t always objects; they are sometimes events, i.e. periodical cycles of events (that’s the only sense in which events can be ‘permanent’). The earth is a permanent cause or primitive natural agent, and so is its rotation. It’s a cause that has always produced. . . .the succession of day and night, the ebb and flow of the sea, and many other effects; and because we can’t assign a cause for the rotation itself (except by guessing!), it is entitled to be classified as a primeval ·or original· cause. But it’s only the origin of the rotation that is mysterious to us; once the rotation has begun, its continuance is accounted for by •the first law of motion (that of the permanence of rectilinear motion once it has been started) combined with •the gravitation of the parts of the earth toward one another. [In that paragraph, ‘produced the succession of day and night’ replaces Mill’s ‘produced (by the aid of other necessary conditions) the succession of day and night’. Throughout §8, almost every statement about x causing y includes a clause about other necessary conditions; these clauses are omitted here in the interests of brevity.]

All phenomena that begin to exist, i.e. all but the primeval causes, are immediate or remote effects those primitive facts or of some combination of them. Throughout the known universe no thing comes into existence and no event happens that isn’t connected by a uniformity or invariable sequence with one or more phenomena that preceded it; so that it will happen again as often as those phenomena occur again and no counteracting cause coexists. These antecedent phenomena were connected in a similar way with some that preceded them,. . . . and so on until we reach our limit, the properties of one or more primeval causes. The whole of the phenomena of nature were therefore the necessary—i.e. the unconditional—consequences of some former collocation [see Glossary] of the permanent causes.

The state of the whole universe at any instant, I believe, is the consequence of its state at the previous instant; so that someone who knew

•all the agents that exist right now,

•their collocations in space, and

•all their properties, i.e. the laws of their agency,

could predict the whole history of the universe from now on, unless some new volition of a power capable of controlling the universe should take over. [How did volition get into the picture? Mill is talking about a change in the basic causal organisation of the universe, and is assuming that if that were to change it would have to be because God—‘a power capable of controlling the universe’—decided to change it.][4] [Mill adds the somewhat isolated remark that if any one total state of the universe came around a second time, the whole history of the universe would repeat itself for ever, ‘like a circulating decimal’. Then he gets back on track:] The whole series of events in the history of the universe, past and future, is intrinsically capable of being constructed a priori by anyone who is acquainted with the original distribution of all natural agents, and with all their properties—i.e. the laws of succession existing between them and their effects. Of course this would require superhuman powers of combination and calculation, even for someone who knew all the original facts.

§9. Because everything that occurs is determined by laws of causation and collocations of the original causes, it follows that coexistences among effects can’t be themselves the subject of any similar set of laws distinct from laws of causation. There are uniformities of coexistence and succession among effects; but these must all result from the identity or the coexistence of their causes; if the causes didn’t coexist, nor could the effects. And these causes being effects of still earlier causes, and these of others,. . . . until we reach the primeval causes, it follows that (apart from effects that can be traced back to a single cause) the coexistences of phenomena can’t be universal unless the coexistences of their primeval causes can be reduced to a universal law; but we have seen that they can’t. So there are no original and independent—i.e. no unconditional—uniformities of coexistence, between effects of different causes; if they coexist, it’s only because their causes have happened to coexist. The only independent and unconditional coexistences that are invariable enough to have a claim to be laws are between different and mutually independent effects of a single cause, i.e. between different properties of the same natural agent. This portion of the laws of nature will be treated of in the chapter 22.2 .

§10. Since the first edition of this work, the sciences of physical nature have made a great advance in generalisation, through the doctrine of the conservation or persistence of force. Building and laying out this imposing edifice of theory has for some time been the principal occupation of the most systematic physicists. It consists of two stages: (1) one consisting of ascertained fact, and (2) one containing, along with some fact, a large element of hypothesis.

(1) It is proved by numerous facts, some experimental, some informal, that agencies that had been regarded as distinct and independent sources of force—heat, electricity, chemical action, nervous and muscular action, momentum of moving bodies—are interchangeable with one another in definite and fixed quantities. It had long been known that these dissimilar phenomena had the power, under certain conditions, to produce one another; what is new in the theory is a more accurate estimation of what this production consists in. What happens is that phenomena of one kind disappear and are replaced by phenomena another kind, and that there is an equivalence in quantity between the phenomena that have disappeared and the ones that have replaced them; so that if the process is reversed, the exact same quantity that had disappeared will re-appear. For example, the amount of heat that will raise the temperature of a pound of water one degree centigrade will, if used in the expansion of steam, lift a weight of 772 pounds one foot, or a weight of one pound 772 feet; and the same quantity of heat can by certain means be recovered through the expenditure of exactly that amount of mechanical motion.

The establishment of this comprehensive law has led the scientific world to change how it speaks about what are called the ‘forces of nature’. Before this correlation between very different phenomena had been discovered, their unalikeness had caused them to regarded as upshots of such-and-such distinct forces. Now that they are known to be convertible into one another without loss, they are spoken of as all the results of one single force, showing itself in different ways. This force (it is said) can only produce a limited and definite quantity of effect, but it always does produce that definite quantity; and produces it (according to circumstances) in one or another of the forms, or divides it among several, but so as always to make up the same sum; and no one of the manifestations can be produced except by the disappearance of the equivalent quantity of another, which in its turn, in appropriate circumstances, will re-appear undiminished. This mutual interchangeability of the forces of nature according to fixed numerical equivalents is the part of the new doctrine that rests on unchallengeable fact. (The judgments about equivalents are based on a scale of numerical equivalents established by experiment.)

An indefinite and perhaps immense interval of time may elapse between the disappearance of the force in one form and its re-appearance in another. A stone thrown up into the air with a given force and falling back immediately will, by the time it reaches the earth, recover the exact amount of mechanical momentum which was expended in throwing it up (minus a small portion of motion given to the air). But if the stone lands on a high ledge it may not fall back for years, perhaps ages, and until then the force used in raising it is temporarily lost, being represented only by what the language of the new theory calls ‘potential energy’. The coal embedded in the earth is considered by the theory as a vast reservoir of force that has remained dormant for many geological periods, and will remain dormant until by being burned it gives out the stored-up force in the form of heat. . . . This means simply that when the coal does at last. . . .generate a quantity of heat (transformable like all other heat into mechanical momentum and the other forms of force), this heat is the re-appearance of a force derived from the sun’s rays, expended myriads of ages ago in the growth of the organic substances that were the material of the coal.

(2) The theory of conservation of force has a part that is a combination of fact and hypothesis. Briefly, it is as follows:

The conservation of force is really the conservation of motion. In each interchange between forms of force it is always motion that is transformed into motion.

This requires the assumption of motions that are hypothetical. The supposition is that •there are molecular motions that appear to our senses only as heat, electricity, etc.— oscillations, invisible to us, among the minute particles of bodies; and that •these molecular motions can be changed into molar motions (motions of masses), and vice versa. We do have positive evidence of the existence of molecular motion in these manifestations of force. In chemical reactions, for instance, the particles separate and form new combinations, often with a great visible disturbance of the mass. And with heat the evidence is equally conclusive, since heat expands bodies (i.e. causes their particles to move apart), and enough heat changes them from solid to liquid, or from liquid to gaseous. Again, the mechanical actions that produce heat—friction, and the collision of bodies—must from the nature of the case produce a shock, i.e. an internal motion of particles, which we often find is so violent as to break them apart. Such facts are thought to justify the conclusion that we were wrong when we thought that heat causes the motion of particles, and that really the motion of particles causes heat; the original cause of both being the earlier motion—molar or molecular, collision of bodies or burning of fuel—that formed the heating agency. This conclusion already contains an hypothesis; but at least the supposed cause, the internal motion of molecules, is a vera causa [see Glossary]. But in order to reduce the conservation of force to conservation of motion, it was necessary to attribute to motion the heat propagated through apparently empty space from the sun. This required the supposition (already made for the explanation of the laws of light) of a superfine ether pervading space; we can’t feel it, but it must have the property that constitutes matter, namely resistance, because waves are propagated through it by an impulse from a given point. This ether must be supposed (and the theory of light doesn’t require this) to penetrate into the minute crevices in all bodies. The story goes like this:

Vibratory motion in the heated mass of the sun is passed on to the particles of the surrounding ether, and through them to the particles of the same ether in the gaps and crevices of terrestrial bodies; and this is done with enough mechanical force to make the particles of those bodies vibrate strongly enough to make the bodies expand and create the sensation of heat in sentient creatures.

This is all hypothesis, but I’m not expressing doubt as to it legitimacy as hypothesis. It seems to follow from this theory that force can and should be defined as matter in motion. But this can’t be right because, as we have seen, the matter doesn’t have to be actually moving. It isn’t necessary to suppose that the motion manifested when the coal burns is actually taking place among the molecules of the coal during its time in the earth;[5] certainly not in the stone at rest on the high ledge onto which it has been thrown. The true definition of force must be potentiality of motion; and what the doctrine. . . .amounts to is not that •there is at all times the same quantity of actual motion in the universe, but that •the possibilities of motion are limited to a definite quantity that can’t be added to and can’t be exhausted; and that all actual motion is a draft on this limited stock [this is a metaphor based on the idea of a bank-draft saying ‘Withdraw £n from account number. . . ’.]. It needn’t all have existed ever as actual motion. There’s a vast amount of potential motion in the universe in the form of gravitation, and it would be a great abuse of hypothesis to suppose that that was stored up by the expenditure of an equal amount of actual motion in some former state of the universe! Nor does the motion produced by gravity take place, as far as we know, at the expense of any other motion of any kind.

If we adopt this theory as a scientific truth, thus accepting its change in our conception of the most general physical agencies, does this require •any change in the view I have taken of causation as a law of nature? As far as I can see, •none whatever. The manifestations that the theory regards as modes of motion are just as distinct and separate phenomena when attributed to •a single force as when attributed to •several. Whether the phenomenon is called a transformation of force or the generation of one, it has its own set(s) of antecedents with which it is connected by invariable and unconditional following; and that set, or those sets, of antecedents are its cause. [Mill now embarks on a long discussion of how he should word his theory in the light of the conservation theory. He refers in friendly terms to a detailed discussion of this by Bain, to whom he attributes the conclusion that:]

In the assemblage of conditions that constitutes the cause of a phenomenon we must distinguish two elements—•the presence of a force and •the collocation or position of objects that is required for the force to undergo the particular transmutation that constitutes the phenomenon.

[Mill accepts this, sort of, but argues at great length that this doesn’t really require him to alter any of his formulations. He accepts and indeed insists that the cause of any change must include a change: ‘To produce a bonfire there must not only be fuel and air and a spark, which are collocations [see Glossary], but chemical action between the air and the materials, which is a force.’ But the insistence on motion, he says, is simply wrong unless we include potential motion; and we must be careful about what we mean by that: ‘The force said to be laid up and merely potential is no more a really existing thing than any other properties of objects are really existing things. The phrase ·“potential force” or “potential motion”· is a mere linguistic device that is convenient for describing the phenomena.’ He concludes:]

We thus see that no new general conception of causation is introduced by the conservation theory. The indestructibility of force doesn’t interfere with the theory of causation any more than the indestructibility of matter does. . . . It only enables us to understand better than before the nature and laws of some of the sequences.

This better understanding, however, lets us join Bain in accepting the expenditure or transfer of energy as one of the tests for distinguishing causation from mere concomitance. If the effect being explained includes matter’s beginning to move, then any of the objects present that has lost motion has contributed to the effect; and this is the true meaning of the thesis that the cause is the one of the antecedents that exerts active force.

§11. This is the place to discuss a rather ancient doctrine about causation that has been revived in recent years and now shows more signs of life than any other theory of causation that conflicts with the one I have been defending.

According to the theory in question, the only cause of phenomena is mind, or more exactly will. The type [see Glossary] of causation, as well as the only source for our idea of it, is our own voluntary agency. The theory’s friends say:

Our voluntary agency is our only direct evidence of causation. We know that we can move our bodies. Regarding the phenomena of inanimate nature, all we have other direct knowledge of is what happens before what. But in our voluntary actions we’re conscious of •power before we have experience of •results. An act of volition, whether or not followed by an effect, is accompanied by a consciousness of effort, ‘of force exerted, of power in action, which is necessarily causal or causative’ [quoted from Francis Bowen]. This feeling of energy or force, inherent in an act of will, is knowledge a priori; assurance before experience that we have the power to cause effects. So volition is more than an •unconditional antecedent; it is a •cause in a different sense of ‘cause’ from that in which physical phenomena are said to cause one another. It is an efficient cause [see Glossary].

It’s easy to move from this to the doctrine that volition is the only efficient cause of all phenomena. ‘It is inconceivable that dead force could continue unsupported for a moment beyond its creation. We cannot even conceive of change or phenomena without the energy of a mind’ [quoted from R. H. Hutton]. And another writer of the same school says: ‘The word “action” itself has no real significance except when applied to the doings of an intelligent agent. Let anyone conceive, if he can, of any power, energy, or force inherent in a lump of matter’ [Bowen again]. Phenomena may seem to be produced by physical causes but they are really produced, say these writers, by the immediate agency of mind. Everything that doesn’t proceed from a human (or, I suppose, an animal) will proceed, they say, directly from divine will. The earth is not moved by the combination of a •centripetal and a •projectile force; this is a mere way of speaking that helps to make our conceptions easier. The earth (they say) is moved by the direct volition of an omnipotent Being, on a path coinciding with the one that we deduce from the hypothesis of •these two forces.

As I have so often said, the general question of the existence of •efficient causes doesn’t fall within the limits of my present subject; but a theory that represents •them as capable of being known by humans, and passes off as efficient causes what are only physical or phenomenal causes, belongs to logic as much as to metaphysics, and is a fit subject for discussion here.

As I see it, a volition isn’t an efficient cause but simply a physical cause. Our will ‘causes’ our bodily actions in exactly and only the sense in which cold causes ice, or a spark causes an explosion of gunpowder. The volition, a state of our mind, is the antecedent; the motion of our limbs in conformity to the volition is the consequent. This sequence is not a subject of direct consciousness in the sense intended by the theory. The antecedent and the consequent are indeed subjects of consciousness; but the connection between them is a subject of experience. Our consciousness of the volition doesn’t contain in itself any a priori knowledge that the muscular motion will follow. If our nerves of motion were paralysed. . . .and had been so all our lives, I don’t see the slightest reason to think that we would ever (unless told by other people) have known anything of volition as a physical power, or been conscious of any tendency in feelings of our mind to produce motions of our body. . . . Would we in that case have had the physical feeling that I suppose these writers mean by ‘consciousness of effort’? I don’t see why not, because that physical feeling is probably a state of nervous sensation beginning and ending in the brain, without involving the motor apparatus; but we certainly wouldn’t have called it anything like ‘effort’. . . . If we were conscious of this sensation, we’d have been conscious of it, I think, only as a kind of uneasiness accompanying our feelings of desire.

Hamilton argues well against the theory in question, thus:

‘It is refuted by the consideration that between •the overt fact of corporeal movement that we know and •the internal act of mental determination that we also know, there intervenes a series of intermediate agencies of which we know nothing; so we can’t be conscious of any causal connection between the volition and the movement, as this hypothesis asserts. . . . A paralytic learns after the volition that his limbs don’t obey his mind; and it’s only after the volition that the healthy man learns that his limbs do obey the mandates of his will.’[6]

Those I am arguing against have never produced, and don’t claim to produce, any positive evidence that the power of our will to move our bodies would be known to us independently of experience.[7] What they say about this is that the production of physical events by a will seems to carry its own explanation with it, while the action of matter on matter seems to require something else to explain it, and is even ‘inconceivable’ unless we suppose that some will intervenes between the apparent cause and its apparent effect. So they base their argument on an appeal to ·what they think to be· the inherent laws of our conceptive faculty, mistaking for •the ·innate· laws of that faculty •its acquired habits based on the spontaneous tendencies of its uncultured state. The sequence from •the will to move a limb and •the actual motion is one of the most direct and instantaneous of all the sequences we observe, and is familiar to every moment’s experience from our earliest infancy. It is more familiar to us than any succession of events exterior to our bodies, and especially more so than any other case of the apparent beginning (as distinguished from the mere passing on) of motion. Our mind naturally tends to be constantly trying to help its conception of unfamiliar facts by assimilating them to familiar ones. And so our voluntary acts. . . .in the infancy and early youth of the human race are spontaneously taken as the type [see Glossary] of causation in general, and all phenomena are supposed to be directly produced by the will of some sentient being. I shan’t describe this primitive idol-worship in the words of Hume or of any of his followers; rather, I’ll take the words of a religious metaphysician, Reid, in order to bring out that all competent thinkers are unanimous on this topic.

·START OF QUOTATION FROM REID·

‘When we turn our attention to external objects and begin to exercise our rational faculties about them, we find that there are some motions and changes in them that we have power to produce, and that many must have some other cause. Either the objects must have life and active power, as we have, or they must be moved or changed by something that has life and active power, as external objects are moved by us.

‘Our first thoughts seem to be that the objects in which we perceive such motions have understanding and active power as we have. “Savages”, says the Abbé Raynal, “wherever they see motion that they can’t account for, postulate a soul.” All men can be considered as “savages” in this respect, until they can be taught and can use their faculties better than savages do. . . .

‘Raynal’s remark is sufficiently confirmed both from fact and from the structure of all languages.

‘Primitive nations really do believe that the sun, moon, and stars, the earth, sea, and air, and fountains and lakes have understanding and active power. Savages find it natural to bow down to these things and beg for their favour, as a kind of idolatry.

‘All languages carry in their structure the marks of their having been formed at a time when this belief prevailed. The division of verbs and participles into •active and •passive, which is found in all languages, must have been originally intended to distinguish what is really active from what is merely passive; and, in all languages we find active verbs applied to the sorts of things in which, according to Raynal, savages think there is a soul.

‘Thus we say “The sun rises and sets”, “The moon changes”, “The sea ebbs and flows”, “The winds blow”. Languages were formed by men who believed these objects to have life and active power in themselves, and so ·for them· it was proper and natural to report such motions and changes with active verbs.

‘There’s no surer way of tracking what nations believed before they had records than by the structure of their language; despite the changes produced in it by time, a language will always bear traces of the thoughts of those who invented it. When we find the same beliefs indicated in the structure of all languages, those beliefs must have been common to the whole human species when languages were being invented.

‘When a few people with superior intellectual abilities find leisure for speculation, they begin to do science [Reid writes: ‘to philosophise’], and they soon discover that many of the things they used to regard as thinking and active are really lifeless and passive. This is a very important discovery. It elevates the mind, frees men from many ignorant superstitions, and opens the door to further discoveries of the same kind.

‘As science advances, life and activity in natural objects retreats, leaving the objects dead and inactive. We find that rather than •moving voluntarily they •are moved necessarily; rather than •acting they are •acted-upon; and nature appears as one great machine in which one wheel is turned by another, that by a third; and the scientist doesn’t know how far back this necessary sequence may reach.’ [Reid, Essays on the Active Powers of Man IV.3.]

·END OF QUOTATION FROM REID·

So there’s a spontaneous tendency of the intellect to explain all cases of causation by assimilating them to the intentional acts of voluntary agents like itself. This is the instinctive philosophy of the human mind in its earliest stage, before it has become familiar with invariable sequences other than those between volitions and voluntary acts. As the notion of fixed laws of succession among external phenomena gradually takes hold, the propensity to explain all phenomena in terms of voluntary agency slowly gives way. But the suggestions of daily life continue to be more powerful than those of scientific thought, so the original instinctive philosophy maintains its ground in the mind, underneath the growths obtained by cultivation [see Glossary], and keeps up a constant resistance to their driving their roots deep into the soil. The theory I’m attacking is fed by that substratum. Its strength lies not in argument but in its link with an obstinate tendency of the infancy of the human mind.

There’s plenty of evidence that this tendency isn’t the result of an inherent mental law. The history of science right from the beginning shows that mankind haven’t been unanimous in thinking either that (i) the action of matter on matter wasn’t conceivable or that (ii) the action of mind on matter was. To some thinkers, ancient and modern, (ii) has seemed much more inconceivable than (i). Sequences that are entirely physical and material, as soon as they became familiar, came to be thought perfectly natural, and were regarded not only as not needing to be explained but as being able to explain other sequences—and even of serving as the ultimate explanation of things in general.

One of the ablest recent supporters of the volitional theory [Hutton] has provided an historically true and philosophically sharp account of the Greek philosophers’ failure in physical inquiry—an account in which, it seems to me, he unconsciously depicts his own state of mind:

‘Their stumbling-block concerned the nature of the evidence they expected for their conviction. . . They hadn’t grasped that they mustn’t expect to understand the •processes of external causes but only their •results; so the whole physical philosophy of the Greeks was an attempt to identify mentally the effect with its cause, to probe for a •connection that was not only necessary but natural—meaning that •it would carry within itself some reason why this antecedent should produce this consequent—and they confined themselves to looking such reasons.’

That is, they weren’t content merely to know that one phenomenon x was always followed by another y; they thought that science’s true aim was to perceive something in x’s nature from which they could have known or presumed previous to trial that it would be followed by y. . . . To complete his statement of the case, the quoted writer should have added that these early speculators not only •had that aim but •thought they had achieved it. The writer can see plainly that this was an error, because he doesn’t believe that any relations between material phenomena can account for their producing one another; but the Greeks’ persistence in this error shows that their minds were in a very different state; the assimilation of physical facts to other physical facts gave them the kind of mental satisfaction that we connect with the word ‘explanation’. . . . When Thales and Hippo held that moisture was the universal cause and external element of which all other things were merely sensible manifestations; when Anaximenes said the same thing about air, Pythagoras about numbers, and the like; they all thought they had found a real explanation and were content to settle for this as ultimate. The ordinary sequences of the external universe seemed to them. . . .to be inconceivable without the supposition of some universal agency to connect the antecedents with the consequents; but they didn’t think that mental volition was the only agency that fulfilled this requirement. Moisture, or air, or numbers, carried to their minds a precisely similar impression of making intelligible what was otherwise inconceivable, and gave the same full satisfaction to the demands of their conceptive faculty.

It wasn’t only the Greeks who ‘wanted to see some reason why the physical antecedent should produce this particular consequent’. . . . Among modern philosophers, Leibniz laid it down as a self-evident principle that all physical causes must contain in their own nature something making it intelligible that they should be able to produce the effects that they do produce. Far from admitting volition as the only kind of cause that carries internal evidence of its own power, and as the real bond of connection between physical antecedents and their consequents, he demanded some naturally and intrinsically efficient [see Glossary] physical antecedent as the bond of connection between volition itself and its effects. He clearly refused to admit the will of God as a sufficient explanation of anything but miracles; and insisted on finding something that would account better for the phenomena of nature than a mere reference to divine volition.

And the action of mind on matter (which, we’re being told, needs no explanation and itself explains all other effects) has seemed to some thinkers to be itself the grand inconceivability. This was the difficulty the Cartesians were trying to solve with the system of ‘occasional causes’. They couldn’t conceive that thoughts in a mind could produce movements in a body, or that bodily movements could produce thoughts. They couldn’t see any necessary connection, any a priori relation, between a motion and a thought. And their insistence—greater than any other philosophical school before or since—that their own minds were measure of all things led them to refuse on principle to believe that Nature had done what they couldn’t see any reason why she must do, so they said it was impossible that a material and a mental fact could be causes one of another. They regarded them as mere ‘occasions’ on which the real agent, God, thought fit to exert his power as a cause. When a man wills to move his foot, they said, it’s not his will that moves it; God moves it on the occasion of the man’s will. And when they looked more carefully into the action of matter on matter they found this inconceivable too, and therefore (according to their logic) impossible. The deus ex machina [see Glossary] was ultimately called in to produce a spark on the occasion of a flint and steel coming together, or to break an egg on the occasion of its falling on the ground.

All this shows that mankind in general is disposed not to be satisfied with knowing that one fact is invariably antecedent and another consequent, but to look for something that may seem to explain their being so. But we also see that this demand can be completely satisfied by a purely physical agency, provided it’s much more familiar than what it is invoked to explain. To Thales and Anaximenes it seemed inconceivable that the antecedents that we see in nature should produce the consequents, but perfectly natural that water or air should produce them. The writers I am opposing in this section declare this to be inconceivable, but they can conceive that mind or volition is an efficient cause; while the Cartesians couldn’t conceive even that, but briskly declared that the only conceivable mode of production of any fact whatever is the direct agency of an omnipotent being; all of which is further evidence for something that finds new confirmation in every stage of the history of science—namely that •what people can conceive and what they can’t is very much an affair of accident, and depends entirely on their experience and their habits of thought; that •by cultivating the required associations of ideas people can make themselves unable to conceive any given thing, and make themselves able to conceive most things, however inconceivable these may at first appear; and that •the facts in each person’s mental history that determine what is or isn’t conceivable to him also determine which sequences in nature will appear to him so natural and plausible as to need no other proof of their existence, and to be evident by their own light independently of experience and of explanation.

By what rule can we decide between one theory of this sort and another? The theorists don’t direct us to any external evidence; each appeals to his own subjective feelings.

One (X) says:

The succession C, B appears to me more natural, conceivable, and intrinsically credible than the succession A, B; so you are wrong in thinking that B depends on A; I am certain—though I can’t give any other evidence of it, that C comes between A and B, and is the real and only cause of B.

Another (Y) answers:

The successions C, B and A, B appear to me equally natural and conceivable, or the latter more so than the former. A is quite capable of producing B without any other intervention.

A third (Z) says:

Like X I can’t conceive that A can produce B; but ·I don’t share his view that C produces B, because· I find the sequence D, B more natural than C, B, so I prefer my D theory to the C theory.

The only universal law operating here is the one saying that each person’s conceptions are governed and limited by his individual experiences and habits of thought! We’re justified in saying of all three, what each of them already believes of the other two, namely that they exalt into an original law of the human intellect and of outward nature one particular sequence of phenomena that they find more natural and more conceivable than other sequences, only because it is more familiar. And I apply this judgment to the theory that volition is an efficient cause.

Before leaving this subject I must mention the additional fallacy contained in the inference from this theory that because volition is an efficient cause, therefore it is the only cause and the direct agent in producing even what is apparently produced by something else. Volitions are not known to produce anything directly except activity in the nerves, because the will influences even the muscles only through the nerves. Suppose we grant that every phenomenon has an efficient [see Glossary] cause and not merely a phenomenal cause, and that volition in the case of the special phenomena that are known to be produced by it is that efficient cause; are we therefore to say (as these writers do) that because we know of no other efficient cause, and oughtn’t to assume one without evidence, there is no other, and volition is the direct cause of all phenomena? [Mill scornfully dismisses this. If our volition is an efficient cause, it’s the only one we can be conscious of because ‘it is the only one that exists within ourselves’; and it’s absurd to infer that volition is the only efficient cause in the universe. Mill likens this to the inference that because we know for sure that there is life on this planet, we can infer that there is life on every heavenly body. He concludes:] I ascribe to certain other creatures a life like my own, because they show the same sort of signs of it as I do. . . . Earth, fire, mountains, trees, are remarkable agencies, but their phenomena don’t conform to the same laws as my actions do, so I don’t attribute animal life to them. But the supporters of the volition theory ask us to infer that volition causes everything simply because it causes one particular thing; although that one thing. . . .is utterly special, its laws being enormously unlike those of any other phenomenon, organic or inorganic.

·SUPPLEMENTARY NOTE TO CHAPTER 5·

[In this densely learned four-page note, Mill responds to critics who accuse him of misrepresenting the views of Thales and Anaximines, and of Descartes and Leibniz. (He points out that he didn’t mention Descartes, only the Cartesians.) Mill’s response opens with all cannons firing: ‘A greater quantity of historical error has seldom been comprised in a single sentence.’ Regarding the ancient philosophers he adduces more evidence, and also shows that his critics’ rival views about Thales and Anaximines are based on misreadings of ancient texts, and ignorance of what Aristotle and others thought about who had had what theory.

[Mill side-tracks at some length into Aristotle’s views about causation in the natural world, mainly so as to highlight two aspects of them. (a) Aristotle held that chance is an efficient cause (though not of everything). We now know that this was an error, Mill writes, but it wasn’t a disreputable one:] Chance had as good a claim to real existence as many other of the mind’s abstract creations; it had been given a name, and why should it not be a reality? (b) The parts of nature that Aristotle regards as representing evidence of design are the uniformities—the phenomena that conform to laws. The common interpretation of nature—we could call it the instinctive, religious interpretation of nature—is the reverse of this. The events in which men spontaneously see the hand of a supernatural being are the ones that they can’t bring under physical laws. Events that they can clearly connect with physical causes, and especially ones they can predict, seem to them not to bear so obviously the mark of a divine will (though they may think that God is responsible for those too). . . . Some eminent writers on natural theology [see Glossary] . . . .think that although design is present everywhere, the irresistible evidence of it is to be found not in the laws of nature but in the collocations [see Glossary], i.e. in the part of nature that shows no signs of any law. A few properties of dead matter might, they think, conceivably account for the regular and invariable succession of effects and causes; but they see proof of a divine providence in the way the different kinds of matter have been so placed as to promote beneficent ends. [Very roughly: It might be possible to explain how your body works without bringing in God, but we have to appeal to God to explain how there comes to be such a material configuration as your body.]

[Mill shows that he was certainly right in what he wrote about the views of Leibniz. We needn’t spend time on this, except to note the tone of anger:] The critics say that what Leibniz found to be inconceivable was not •that mind acts on matter but •how it does so. This is an abuse of the privilege of writing confidently about authors without reading them! If my critics knew anything about Leibniz they would know that for him ‘the inconceivability of how’ and ‘the impossibility of the thing’ were equivalent expressions. . . .

[Regarding the Cartesians, Mill focuses on Malebranche, the best known Cartesian and the chief expositor (though not the inventor) of ‘the system of occasional causes’, and easily shows that he (Mill) was right about his views. He concludes: ‘If Malebranche hadn’t believed in an omnipotent Being, he would have held all action of mind on body to be a demonstrated impossibility.’

[There’s a further half-page of tussle with the critics, but we can safely by-pass this efficient operation of garbage-removal.]

NOTES

[1] There are a few ·apparent· exceptions; for some properties of objects seem to be purely preventive, e.g. the property of opaque bodies by which they intercept the passage of light. This, as far as we can understand it, seems to be a case of an agency that shows up only in defeating the effects of another agency. If we knew what other relations to light, or what peculiarities of structure, opacity depends on, we might find that this is only an apparent exception to the general proposition in the text, not a real one. Either way, it needn’t affect the practical application. The formula that includes all the negative conditions of an effect in the single one of the absence of counteracting causes is not violated by such cases as this; though if all counteracting agencies were like this there would be no point in employing the formula.

[2] By ‘the present constitution of things’ I mean the ultimate laws of nature (whatever they may be), as distinct from the derivative laws and from the collocations [see Glossary]. The daily revolution of the earth (for example) is not a part of the constitution of things, because it could be terminated or altered by natural causes.

[3] I say ‘straight line’ for brevity and simplicity. In reality the line in question is not exactly straight: because of refraction, we actually see the sun for a short interval during which the opaque mass of the earth is interposed in a direct line between the sun and our eyes. This provides us with a limited version of the luxury of seeing round a corner.

[4] In this footnote Mill mentions those who think that human volition constitutes an exception to the determinist thesis that whatever happens is caused to happen. He says that he’ll deal with this thoroughly in VI.2, and right now will make just one point:] These metaphysicians base their objection ·to determinism· on the claim that it conflicts with our consciousness. I think they mistake the proposition that consciousness testifies against. If they look into themselves carefully, they’ll find that what their consciousness objects to is the thesis that human actions and volitions are necessary in the everyday sense of that term. I agree with them about that. But the statement ‘A person’s actions necessarily follow from his character’ really means what is meant in any statement about causation, namely that •the person invariably does act in conformity to his character, and •that anyone who thoroughly knew his character could predict with certainty how he would act in any supposable case. They probably wouldn’t find this contrary to their experience or revolting to their feelings. And no-one claims more than this, except for Asiatic fatalists.

[5] I understand that the accredited authorities do suppose that molecular motion, equivalent in amount to what will be manifested when the coal burns, is actually taking place during the whole of that long interval, not in the coal but in the oxygen that will then combine with it. You can see how purely hypothetical this supposition is; and I venture to say that it is unnecessarily and extravagantly hypothetical.

[6] . . . .This acute thinker has a theory of causation that is all his own. As far as I know it has never been analytically examined, but I think it can be refuted as completely any one of the false or insufficient psychological theories that strew the ground under his potent metaphysical scythe. (Since writing that I have examined and controverted his theory in my Examination of Sir William Hamilton’s Philosophy, ch. 16.)

[7] Bowen disagrees: ‘The result to be accomplished is preconsidered or meditated, and is therefore known a priori or before experience.’ This merely says that when we will a thing we have an idea of it. . . .but that doesn’t imply a prophetic knowledge that it will happen. You may object: ‘The first time we exerted our will, when we had no experience of any of our powers, we must have known that we had those powers, because we can’t will something that we don’t believe to be in our power.’ But that’s a merely verbal impossibility. We can desire something that we don’t know to be in our power, and when we find by experience that our bodies move according to our desire then we can pass into the more complicated mental state that is termed will. . . .

Chapter 6. The composition of causes

§1. To complete the general notion of causation. . . ., one distinction still remains to be pointed out. It is so radical and so important that it requires a chapter to itself.

We’re now familiar with the case in which several agents or causes jointly produce an effect. It is indeed the usual case: very few effects are produced by just one agent. Suppose that two agents, operating jointly, are followed (under certain collateral conditions) by a given effect. If either of them had operated alone (under those same conditions), some effect would probably have followed, an effect •different from the joint effect of the two and •more or less dissimilar to it. When we know what would be the effect of each cause acting alone, we can often arrive deductively—i.e. a priori—at a correct prediction of what will arise from their joint agency. We can do this just so long as

the law expressing the effect of each cause acting by itself also correctly expresses that cause’s part of the effect that follows from the two together.

That’s how things stand in the important class of phenomena commonly called ‘mechanical’, namely the phenomena of the communication of •motion from one body to another (or of •pressure, which is tendency to motion). In this class of cases it never happens that one cause defeats or frustrates another; both have their full effect. If a body is propelled by one force tending to drive it to the north and by another to the east, it is caused to move in a given time exactly as far in both directions as the two forces would separately have carried it. It ends up precisely where it would have arrived if it had been acted on first by one of the two forces and then by the other. In dynamics this law of nature is called the principle of the composition of forces; and in imitation of that well-chosen label I shall give the name ‘composition of causes’ to the principle that is exemplified whenever the joint effect of several causes is identical with the sum of their separate effects.

This principle doesn’t prevail in all parts of nature. The chemical combination of two substances produces, as is well known, a third substance with properties different from those of either of the two substances separately. . . . No trace of the properties of hydrogen or of oxygen is observable in their compound, water. The taste of lead acetate isn’t the sum of the tastes of its component elements, acetic acid and lead or its oxide; nor is the colour of blue vitriol a mixture of the colours of sulphuric acid and of copper. This explains why mechanics is a deductive or demonstrative science, and chemistry is not. In mechanics we can compute the effects of combinations of causes from the laws that we know to govern those causes when acting separately, because they conform to the same laws when in combination that they conform to when acting separately. . . . Not so in the phenomena that are the special subject of the science of chemistry. There most of the uniformities the causes conform to when separate cease altogether when they are conjoined, and we can’t (at least in the present state of our knowledge) foresee what result will follow from any new combination until we have tried the specific experiment.

If this is true of chemical combinations, it’s even more true of the far more complex combinations of elements that constitute organised bodies—combinations out of which arise the extraordinary new uniformities called the ‘laws of life’. All the parts of organised bodies are similar to the parts of inorganic things, and have themselves existed in an inorganic state; but the phenomena of life that result from the juxtaposition of those parts in a certain manner are utterly unlike all the effects that would be produced by the action of the component substances acting as mere physical agents. No imaginable knowledge of a living body’s ingredients, however wide-ranging and complete, could enable us to predict the events of the living body itself from our knowledge of the separate actions of its elements. The tongue, for instance, is composed of gelatine, fibrin, and other products of the chemistry of digestion; but from no knowledge of the properties of those substances could we ever predict that the tongue could taste, unless gelatine or fibrine could themselves taste, for no elementary fact can be in the conclusion that wasn’t in the premises.

Thus the combined action of several causes can belong to either of two types, from which arise two ways in which laws of nature can conflict or interfere with one another. Take a case where at a given point of time and space there are two or more causes which, if they acted separately, would produce effects contrary to—or at least conflicting with—each other, one of them tending to undo some or all of what the other tends to do. Examples:

•The expansive force of the gases generated by the ignition of gunpowder tends to launch a bullet toward the sky, while its gravity tends to make it fall to the ground.

•A stream running into a reservoir at one end tends to raise its level higher and higher, while a drain at the other end tends to empty it.

In cases such as these, although the two causes exactly annul one another the laws of both are still fulfilled; the effect is the same as if the drain had been open for half an hour first, and the stream had flowed in for half an hour afterward.[1] Each agent produces the same amount of effect as if it had acted separately, though the contrary effect occurring at the same time obliterated it as fast as it was produced. Here, then, are two causes producing by their joint operations an effect that •at first seems quite unlike the effects they produce separately but •on examination proves to be really the sum of those separate effects. . . .

So there’s one kind of mutual interference of laws of nature in which, even when the joint causes annihilate each other’s effects, each exerts its full efficacy according to its own law as a separate agent. In the other kind of case, the agencies that are brought together cease entirely, and a totally different set of phenomena arise—e.g. when two liquids are mixed in certain proportions they instantly become not a larger amount of liquid but a solid mass.

§2. This difference between. . . .•laws that work together without alteration and •laws which, when called on to work together, cease and give place to others, is one of the fundamental distinctions in nature. The former case (the composition of causes) is the usual one; the other is always special and exceptional. There are no objects that don’t obey the principle of the composition of causes with regard to some of their effects. For instance, a body retains it weight in all the combinations in which it is placed. The weight of a chemical compound, or of an organised body, is equal to the sum of the weights of the elements composing it. The weight will be lessened if the body is moved further from the centre of attraction, but it will be the same lessening for the compound as for the elements. The component parts of a plant or animal don’t lose their mechanical and chemical properties as separate agents when they are spatially inter-related in the special way such that they, as an aggregate whole, acquire physiological or vital properties in addition. Those bodies still obey mechanical and chemical laws, because the operation of those laws isn’t counteracted by the new laws that govern the bodies as organised beings. To put that in another way: when two or more causes jointly operate in a way that calls into action new laws with no resemblance to any we can find in the separate operation of the causes, the new laws, while superseding one portion of the previous laws, may coexist with another portion, and may even compound the effect of those previous laws with their own.

Also, laws that were themselves generated in the second way may generate others in the first. The laws of chemistry and physiology (for example) owe their existence to a breach of the principle of composition of causes, but these heteropathic laws, as we might call them, are capable of composition with one another. The causes which by one combination had their laws altered may carry their new laws with them unaltered into further combinations. So we needn’t despair of eventually raising chemistry and physiology to the condition of deductive sciences; for though it’s impossible to deduce all chemical and physiological truths from the laws or properties of simple substances or elementary agents, perhaps they are deducible from laws that come into play when these elementary agents are brought together into some moderate number of not very complex combinations. The laws of life will never be deducible from the mere laws of the ingredients, but the prodigiously complex facts of life may all be deducible from comparatively simple laws of life—which do indeed depend on combinations, but comparatively simple ones. These laws of life may in more complex circumstances be strictly compounded with one another and with the physical and chemical laws of the ingredients. We already know enough about the vital phenomena to know of countless cases where they enter into the composition of causes; and the more precisely we study these phenomena the more reason we seem to get for believing that the laws that operate in the simpler combinations of circumstances do in fact continue to be observed in more complex ones. This will be found equally true in the phenomena of mind; and even in social and political phenomena, which are results of the laws of mind. It’s with chemical phenomena that the least progress has been made, so far, in bringing the special laws under general ones from which they can be deduced; but even in chemistry there are many circumstances to encourage the hope that such general laws will eventually be discovered. There’s no chance that the different actions of a chemical compound will ever be found to be the sums of the actions of its separate elements; but between the properties of •the compound and those of •its elements there may be some constant relation that would enable us to foresee the sort of compound that will result from a new combination before we have actually tried it, and to judge what sort of elements some new substance is compounded of before we have analysed it. (The relation would of course have to be discovered by a sufficient induction.) The law of definite proportions, first presented in its full generality by [John] Dalton ·in his atomic theory· is a complete solution of this problem so far as •quantity is concerned; and for •quality we already have some partial generalisations suggesting that we may eventually get further. We can know in advance some properties of the kind of compounds that result from combining, in each of the small number of possible proportions, any acid with any base. We also have the curious law discovered by Berthollet: two soluble salts mutually decompose one another whenever the new combinations that result produce a compound that is less soluble than either of the original two. . . . Thus it appears that even heteropathic laws—laws of combined agency that aren’t derived •by simple addition from the laws of the separate agencies—are in some cases derived •according to some fixed principle from the separate laws. So there may be laws governing the generation of laws from others that are unlike them; and in chemistry these undiscovered laws of the dependence of a compound’s properties on the properties of its elements may, together with the laws of the elements themselves, provide the premises by which chemistry is perhaps destined eventually to be made deductive.

So it seems •that the composition of causes occurs in every class of phenomena; •that as a general rule causes in combination produce exactly the same effects as when acting singly; but •that this rule, though general, isn’t universal because in some instances, at some particular points in the transition from separate to united action, the laws change and an entirely new set of effects occur in place of (or in addition to) the effects arising from the separate agency of those same causes; and •that the laws of these new effects are in their turn capable of composition. . . .

§3. That effects are proportional to their causes is laid down by some writers as an axiom in the theory of causation. It has been worked hard in reasonings about the laws of nature, though it is burdened with many difficulties and apparent exceptions which much ingenuity has been expended in showing not to be real ones. What truth there is in this ‘axiom’ is just a special case of the composition of causes— the case where compounded causes are homogeneous, so that their joint effect might be expected to be the sum of their separate effects. ·The ‘axiom’ is illustrated by this·:

A force equal to 100lb will raise a certain body a certain distance along an inclined plane; a force equal to 200lb will raise two ·such· bodies the exact same distances; so the effect is proportional to the cause.

But the 200lb force contains two forces each equal to 100lb— forces each of which would raise one of the bodies if it were employed separately. So the fact used to illustrate the ‘axiom’ results from the composition of causes; it’s a mere instance of the general fact that mechanical forces are subject to the law of composition. And it’s the same in every other conceivable case. The doctrine of the proportionality of effects to their causes obviously can’t apply to cases where adding something to the cause alters the kind of effect. . . . Suppose that the application of a certain quantity of heat to a body merely •increases its size, that a double quantity •melts it, and a triple quantity •decomposes it: because these three effects are heterogeneous, there can’t be any ratio between them, let alone one that matches the ratio among the quantities of heat applied. Thus the ‘axiom’ of the proportionality of effects to their causes fails at the precise point where the principle of the composition of causes fails. . . .

This is the end of my general remarks on causation, which I thought were needed as an introduction to the theory of the inductive process—a process that is essentially an inquiry into cases of causation. All the uniformities in the •succession of phenomena, and most of the uniformities in their •coexistence, are either laws of causation or consequences of such laws. If we could determine what causes are correctly assigned to what effects, and what effects to what causes, that would virtually amount to knowing the whole course of nature. All the uniformities that are mere results of causation might then be explained, and every individual fact or event might be predicted, provided we knew the relevant facts about the circumstances that preceded it.

To ascertain, therefore, what laws of causation there are in nature—to determine the effect of every cause, and the causes of all effects—is the main business of induction. And the chief object of inductive logic is to point out how this is done.

NOTES

[1] Strictly speaking, in the second case the draining would be a little slower because there would be less pressure to create it, but that doesn’t affect the truth of what I’m saying, because that would involve a change in the conditions under which the drain was acting.

Chapter 7. Observation and experiment

§1. One upshot of what I have been saying is that the process of ascertaining •what consequents are invariably connected with what antecedents, i.e. •what phenomena are related to each other as causes and effects, is a sort of process of analysis. We can take it as certain that every fact that begins to exist has a cause, and that this cause is some fact (or facts) that immediately preceded it. The totality of present facts is the infallible result of the totality of past facts, and more immediately of all the facts that existed a moment ago. So we have here a great sequence that we know to be uniform: if the whole moment-ago state of the entire universe could occur again, it would again be followed by the present state. How, then, are we to resolve [see Glossary] this complex uniformity into the simpler uniformities that compose it, and assign to each part of the vast antecedent the part of the consequent that comes from it?

This operation, which I have called ‘analytical’ because it’s the resolution of a complex whole into its component elements, is more than a merely mental analysis. We shan’t get what we want merely by thinking about the phenomena, dividing them by the intellect alone. But such a mental partition is an indispensable first step. At first glance the order of nature looks at every moment like a chaos followed by another chaos! We must decompose each chaos into single facts. We must learn to see in the chaotic antecedent a multitude of distinct antecedents, in the chaotic consequent a multitude of distinct consequents. But this won’t tell us which of the antecedents produces each consequent. To determine that we must try to separate the facts from one another, not only in our minds but in nature. The mental analysis, however, must be done first. And we all know that intellects differ immensely in how they do this. It is of the essence of the act of observing; because the observer doesn’t merely see the thing before his eyes but sees what parts it is composed of. The ability to do this well is rare:

•one person, from inattention or attending only in the wrong place, overlooks half of what he sees;

•another sets down much more than he sees, mixing it up with what he imagines or infers;

•a third takes note of the kind of all the circumstances, but because he’s inexpert in estimating their degree he leaves the quantity of each vague and uncertain;

•a fourth sees the whole, but makes such an awkward division of it into parts—throwing into one mass things that should be separated, and separating others that would be better considered as one—that the result is no better, perhaps worse, than if he hadn’t attempted any analysis.

We might discuss what qualities of mind and kinds of mental culture equip someone to be a good observer; but that belongs not logic but to the theory of education in the broadest sense of that term. There’s no art of observing if ‘art’ is being used properly. There may be rules for observing. But these—like rules for inventing, are really instructions for how to put one’s own mind into the state in which it will be most fitted to observe, or most likely to invent. So they are essentially rules of self-education, which is different from logic. They don’t teach how to do the thing but how to make ourselves capable of doing it. They’re an art of strengthening the limbs, not of using them.

How wide and how detailed does the observation have to be? How far down do we have to go in the mental analysis? That depends on the purpose in view. To ascertain the state of the whole universe at any moment is impossible, and would also be useless. When making chemical experiments we don’t think it necessary to note the position of the planets, because experience has shown. . . .that in such cases that detail isn’t relevant to the result. Thus, at times when men believed in the occult influences of the heavenly bodies it might have been unphilosophical [here = ‘unscientific’] to fail to check on the precise condition of those bodies at the moment of the experiment. As for the degree of minuteness of the mental subdivision, if we had to break down what we observe into •its very simplest elements. . . ., it would be hard to say where we would find •them; we can hardly ever affirm that our divisions of any kind have reached the ultimate unit. But fortunately this doesn’t matter either. The point of the mental separation is to suggest the required physical separation, as something to be done by us or sought for in nature; and we needn’t go beyond the point at which we can see what observations or experiments we require. What does matter is this: at whatever point our mental decomposition of facts has stopped, we should be ready and able to carry it further if there’s a need for that, not allowing the freedom of our discriminating faculty [= ‘our ability to make distinctions’] to be imprisoned by the straps and bindings of ordinary classification. That’s what happened with all early speculative inquirers, the Greeks included. It seldom occurred to them that something called by •one abstract name might actually be •several phenomena, or that the facts of the universe might be decomposable into elements other than the ones already recognised in ordinary language.

§2. Suppose that we have ascertained the different antecedents and consequents and have discriminated them from one another as far as the case requires, we now face the question: Which is connected with which? There are always many antecedents and many consequents. If the antecedents couldn’t be separated from one another except in thought, or if the consequents were never found apart, it would be impossible for us to distinguish the real laws empirically, or to assign to any cause its effect, or to any effect its cause. To do that we have to encounter some of the antecedents apart from the rest, and observe what follows from them; or some of the consequents, and observe what they are preceded by. In short, we must follow the Baconian rule of varying the circumstances. This is indeed only the first rule of physical inquiry, and not the sole rule, as some have thought; but it is the foundation of all the rest.

If we want to vary the circumstances, we can rely on •observation or •experiment; we can either •find in nature an instance suited to our purposes, or •make one by an artificial arrangement of circumstances. The value of the instance depends on what it is in itself, not on how it is obtained; its role in induction depends on the same principles in each case, just as the uses of money are the same whether it is inherited or earned. So there’s no difference in kind, no real logical distinction, between the two processes of investigation. But there are practical differences that it’s important not to overlook.

§3. The most obvious difference is that experiment is an immense extension of observation. As well as enabling us to produce many more variations in the circumstances than nature spontaneously offers, it also (in thousands of cases) enables us to produce just exactly the sort of variation we need for discovering the law of the phenomenon ·we are studying·. Nature is seldom so friendly as to give us that, because it’s not constructed on a plan of helping us to study it!

For example, in order to ascertain what principle [see Glossary] in the atmosphere enables it to sustain life, we need a living animal to be immersed in each component element of the atmosphere separately. But nature doesn’t supply either oxygen or nitrogen in a separate state. We are indebted to artificial experiments for our knowledge that it’s oxygen, not nitrogen, that supports respiration; and for our knowledge of the very existence of those two ingredients.

Everyone realises that experimentation has the advantage over simple observation that it enables us to •obtain ever so many combinations of circumstances that aren’t found in nature, and so •add to nature’s experiments a multitude of experiments of our own. But many people don’t realise that there’s another superiority. . . .of artificially obtained instances over spontaneous ones—of our own experiments over even the same experiments when made by nature—which is at least as important.

When we produce a phenomenon artificially, we can (as it were) take it home with us, and observe it in circumstances [see Glossary] that we know about in detail. When we want to know what the effects are of the cause A, if we can produce A by means at our disposal we can generally determine at our own discretion. . . .the whole of the circumstances that are present along with it; and because this lets us know exactly the simultaneous state of everything else that could interact with A, we have only to observe what alteration is made in that state by the presence of A.

For example the electric machine lets us produce in thoroughly known circumstances the phenomena that nature displays on a grander scale in the form of lightning and thunder. Think about it: How much could mankind have learned about the effects and laws of electric agency from the mere observation of thunder-storms? And compare that with what they have learned and may expect to learn from electrical and galvanic experiments! What makes this example especially striking is the fact—as we now have reason to believe—that electric action is of all natural phenomena (except heat) the most pervading and universal. This might lead you to think that electricity has the least need to be artificially produced in order to be studied; but the fact is the reverse of that—without the electric machine, the Leyden jar, and the voltaic battery we would probably never have suspected the existence of electricity as one of the great agents in nature; the few electric phenomena we would have known of would have gone on being regarded either as supernatural or as a sort of anomaly, an eccentricity in the order of the universe.

When we have insulated the phenomenon we’re investigating by placing it among known circumstances, we can vary the circumstances in any way we like, choosing the variations that we think have the best chance of bringing the laws of the phenomenon into a clear light. By introducing one well-defined circumstance after another into the experiment, we discover how the phenomenon behaves in an indefinite variety of possible circumstances. Thus, when chemists have obtained some newly-discovered substance in a pure state. . . .they •introduce various other substances one by one, to discover whether it will combine with them or decompose them, and with what result; and also •apply heat, or electricity, or pressure, to discover what will happen to the substance in each of these circumstances.

But if we can’t produce the phenomenon, and have to look for occurrences of it in nature, the task before us is very different.

Rather than choosing what the concomitant [see Glossary] circumstances shall be, we now have to discover what they are; and it’s next to impossible to do this with any precision and completeness except in the simplest and most accessible cases. Here’s a phenomenon that we have no means of making artificially—a human mind. Nature produces many; but because we can’t produce them by art we can see a human mind developing or acting on other things only when it is surrounded and obscured by an indefinite multitude of undiscoverable circumstances, making the use of ordinary experimental methods almost delusive. To get a sense of the scope of this difficulty, consider the fact that whenever nature produces a human mind she produces a body closely connected with it; i.e. a vast complex of physical facts, with no two of these complexes being exactly alike (probably), and most of them being radically out of the reach of our means of exploration (except for the mere structure, which we can examine in a coarse way after it has ceased to act [i.e. in an autopsy]). And if instead of a human mind we try to investigate a human society or a state, we encounter all the same difficulties—the same only worse.

We are now within sight of a conclusion that later chapters will (I think) make shiningly evident: in the sciences dealing with phenomena in which •artificial experiments are impossible (such as astronomy), or in which •they have a very limited range (as in psychology, social science, and even physiology), induction from direct experience is practised at a disadvantage that usually amounts to impossibility. If those sciences are to learn anything worth learning, therefore, their methods must be largely and perhaps principally deductive. This is already known to be the case with astronomy; that it’s not generally recognised as true of the others is probably one reason why they aren’t in a more advanced state.

§4. Although pure ·hands-off· observation is at a great disadvantage compared with artificial experimentation in one branch of the direct exploration of phenomena, there’s another branch where the advantage is all on the other side.

Because inductive inquiry aims to learn what causes are connected with what effects, we can begin this search at either end of the road: we can inquire either into •the effects of a given cause or into •the causes of a given effect. The fact that light blackens silver chloride could have been discovered either

(a) by experiments on light, trying what effect it would have on various substances, or

(b) by observing that portions of the chloride had repeatedly become black, and investigating the circumstances.

The effect of the poison curare could have become known either

(a) by administering it to animals, or

(b) by examining how it came about that the wounds the Indians of Guiana inflict with their arrows are always fatal.

A quick look at those examples, with no need for theoretical discussion, shows that artificial experimentation is possible only with the (a) procedures. We can take a cause and try what it will produce; we can’t take an effect and try what it will be produced by. We can only watch till we see it produced, or are enabled to produce it by accident.

This wouldn’t matter much if it was always up to us to choose which end to start from. But we seldom have any option. We can only travel from the known to the unknown, so we have to start at whichever end we know most about. If the agent is more familiar to us than its effects, we watch for or contrive instances of the agent in whatever varieties of circumstances we can manage, and observe the result. If the conditions on which a phenomenon depends are obscure but the phenomenon itself is familiar, we must start our inquiry from the effect. If we’re struck with the fact that silver chloride has been blackened, and have no idea of the cause, all we can do is to compare instances where the blackening has happened to occur, until through that comparison we discover that in all those instances the substances had been exposed to light. If we knew nothing of the Indian arrows but their fatal effect, accident alone could turn our attention to experiments on the poison; in the regular course of investigation we could only investigate or try to observe what had been done to the arrows in particular instances.

Whenever we have no leads on the cause and therefore have to start from the effect and apply the ‘varying the circumstances’ rule to the consequents, not the antecedents, we’re deprived of the resource of artificial experimentation. But this is a matter of looking for or waiting for cases of the consequent in varying circumstances; we can’t produce them because the only way to produce an effect is through its cause, and we don’t know the cause. . . . If nature happens to present us with instances sufficiently varied in their circumstances, and if we can discover something that is always found—either immediately before the effect or some distance further back—when the effect is found and never found when it isn’t, we can discover by mere observation and without experiment a real uniformity in nature.

But although this is certainly the most favourable case for •sciences of pure observation, as contrasted with •sciences in which artificial experiments are possible, there’s really no case that more strikingly illustrates the inherent imperfection of direct induction when not based on experiment. Suppose that by comparing cases of the effect y we find an antecedent x that appears to be invariably connected with it; we haven’t proved x to be the cause of y until we have reversed the process and used x to produce y. If we can produce x artificially, and if when we do so y follows, the induction is complete: we know that antecedent x is the cause of that consequent y.[1] But we got there by adding the evidence of experiment to that of simple observation. [Mill then goes through it all again, with different words but the same content. He sums up:] In short, observation without experiment (and with no aid from deduction) can discover sequences and coexistences but can’t prove causation.

[Mill cites zoology as a science in which an enormous amount is known about what follows what and what coexists with what, and yet:] on this vast subject. . . .we have made most scanty progress in discovering any laws of causation. In most of the cases of coexistence of animal phenomena we don’t know for sure which is the cause and which the effect (or whether they aren’t related as cause and effect but rather are two effects of causes yet to be discovered, complex results of laws hitherto unknown.

Some of what I have said really belongs later, but I thought that a few general remarks on •how sciences of mere observation differ from sciences of experimentation, and on •the extreme disadvantage that inductive inquiry labours under in the former, would be the best preparation for discussing the methods of direct induction. . . ., a discussion to which I now proceed.

NOTES

[1] Unless y was generated not by the x but by the means used to produce the x. But these means are •under our power, so there’s some probability that they are also sufficiently •within our knowledge to enable us to judge whether that could be the case.

Chapter 8. The four methods of experimental inquiry

§1. The aim is to single out from among the circumstances that precede or follow a phenomenon the ones that it is really connected with by an invariable law. Two ways of doing this are simpler and more obvious than any other others. In the Method of Agreementwe compare different instances in which the phenomenon occurs. In the Method of Differencewe compare instances in which it occurs with instances in other respects similar in which it doesn’t.

In illustrating these methods. . . ., I’ll attend to their use both in •inquiring into the cause of a given effect and •inquiring into the effects or properties of a given cause. . . . I’ll denote antecedents by upper-case letters and the corresponding consequents by ·italicised· lower-case.

Let A be an agent or cause, and suppose we are trying to ascertain what its effects are. If we can find or produce A in such varieties of circumstances that the different cases have no circumstance in common except A, then any effect that we find to be produced in all our trials is shown to be the effect of A. Suppose, for example, that A is tried along with B and C, and that the effect is a b c; and suppose that A is next tried with D and E, but without B and C, and that the effect is a d e. Then we may reason thus: b and c are not effects of A, for they weren’t produced by it in the second experiment; nor are d and e, for they weren’t produced in the first. Whatever is really the effect of A must have been produced in both instances, and the only circumstance that’s true of is a. . . .

For example, let the A be the contact of an alkaline substance and an oil. We try this combination in several varieties of circumstances that resemble each other only in that they all produce a greasy and soap-like substance; so we conclude that the combination of an oil and an alkali causes the production of a soap. That is how we use the Method of Agreement to inquire into the effect of a given cause.

In a similar way we can inquire into the cause of a given effect. Let a be the effect. Here. . . .we have only the resource of observation without experiment; we can’t take a phenomenon of which we don’t know the origin and try to find how it is produced by producing it!. . . . But if we can observe a in two different combinations, a b c and a d e; and if we know or can discover that the antecedent circumstances in these cases respectively were A B C and A D E, we can conclude by a reasoning similar to that in the ‘soap’ example that A is the antecedent connected with the consequent a by a law of causation. B and C can’t be causes of a because on its second occurrence they weren’t present; nor can D and E, because they weren’t present on its first occurrence. A is the only one of the five circumstances that was found among the antecedents of a in both instances.

For example, suppose the effect whose cause we want to discover is crystallisation. We compare cases where bodies are known to acquire crystalline structure but have nothing else in common. We find them to have one—and as far as we can see only one—antecedent in common, namely the deposition of a solid matter from a liquid state. . . . So we conclude that the solidification of a substance from a liquid state is an invariable antecedent of its crystallisation.

In this example we can go further and say that this is not only •the invariable antecedent of crystallisation but •the cause of it; or at least the immediately preceding event that completes the cause. That’s because after detecting the antecedent A we can •produce it artificially, and by finding that a follows it •verify the result of our induction. [Mill cites two examples, discoveries about how to produce quartz and how to produce marble. He comments that these are] two admirable examples of the light that can be thrown upon the most secret processes of Nature by well-contrived interrogation of her.

But if we can’t artificially produce A, the conclusion that it’s the cause of a remains very doubtful. Even if it’s an invariable antecedent of a, preceding it as day precedes night, it may not be unconditionally so.

This uncertainty arises from our inability to be sure that A is the only immediate antecedent common to both the instances. If we could be certain of having ascertained all the invariable antecedents, we might be sure that the unconditional invariable antecedent—i.e. the cause—must be among them. Unfortunately it’s hardly ever possible to ascertain all the antecedents unless the phenomenon is one we can produce artificially. Even then, the difficulty is merely lightened, not removed; men knew how to raise water in pumps long before they learned what was really the operating circumstance in pumping, namely the pressure of the atmosphere on the open surface of the water. It’s much easier to analyse completely •a set of arrangements made by ourselves than •the whole complex mass of agencies that nature happens to be exerting at the moment when a given phenomenon is produced. We may overlook some of the relevant circumstances in an experiment with an electrical machine; but at worst we’ll be better acquainted with them than with the circumstances of a thunder-storm.

The way of discovering and proving laws of nature that I have just presented is based on the following axiom:

Whatever circumstances can be absent when the phenomenon is present is not causally connected with it. With such casual circumstances set aside, if only one remains then it is the cause we are searching for; if more than one remains, they either are the cause or contain it among them; and the same thing holds mutatis mutandis [see Glossary] for the effect.

As this method proceeds by comparing different instances to ascertain what they agree in, I call it the Method of Agreement, and we can adopt as its regulating principle the following:

FIRST CANON.

If two or more instances of the phenomenon under investigation have only one circumstance in common, that one circumstance is the cause (or effect) of the given phenomenon.

I’ll return to the Method of Agreement very soon, but first I proceed to a still more powerful instrument in the investigation of nature, the Method of Difference.

§2. The Method of Agreement required instances that agreed in the given circumstance but differed in every other; the present method requires two instances that resemble one another in every other respect but differ in the presence of absence of the phenomenon we wish to study ·and, presumably in the presence or absence of the cause of that phenomenon·. [That addition is needed to save Mill’s account from incoherence. Other instances of the same trouble will be left untreated.] If we’re trying to discover the effects of an agent A, we must procure A in some set of known circumstances A B C, note the effects of that, and compare them with the effect of the remaining circumstances B C without A. If the effect of A B C is a b c, and the effect of B C is b c, it is evident that the effect of A is a. And if we begin at the other end, wanting to investigate the cause of an effect a, we must select an instance a b c in which the effect occurs and the antecedents were A B C, and then look for another instance in which b c occur without a. If in that instance the antecedents are B C, we know that the cause of a must be A—either alone or in conjunction with some other circumstances present.

It’s scarcely necessary to give examples of a logical process that gives us almost all the inductive conclusions we draw in daily life. When a man is shot through the heart, the Method of Difference shows us that it was the gunshot that killed him: he was in the fullness of life immediately before, all circumstances being the same ·as after· except the wound.

The axioms implied in this method are evidently the following. An antecedent that can’t be excluded without preventing the phenomenon is the cause of that phenomenon or a condition of it; a consequent that can be excluded with no other difference in the antecedents than the absence of a particular one x is the effect of x. Instead of comparing different instances of a phenomenon to see how they agree, this method compares an instance of its occurrence with an instance of its non-occurrence to see how they differ. The regulating principle of the Method of Difference may be expressed thus:

SECOND CANON.

If an instance where the phenomenon y under investigation occurs and an instance where it doesn’t occur have every circumstance in common except for one x that occurs only in the former, x is the effect or the cause or an indispensable part of the cause of y.

§3. The two methods I have presented are alike in many ways but also unalike in many way. Both are methods of elimination. This term (borrowed from the mathematical theory of equations. . . .) is well suited to express the operation that has been understood since the time of Bacon to be the foundation of experimental inquiry—namely the successive exclusion of the various circumstances that are found to accompany a phenomenon in a given instance, in order to ascertain which of them can be absent consistently with the existence of the phenomenon. The Method of Agreement is based on the thesis that whatever can be eliminated is not connected with the phenomenon by any law. The Method of Difference has for its foundation that whatever can’t be eliminated is connected with the phenomenon by a law.

Of these two, the Method of Difference is more particularly a method of artificial experiment; while the Method of Agreement is more especially what we use when experimentation is impossible. A few reflections will prove this, and point out the reason of it.

It is inherent in the unique character of the Method of Difference that the nature of the combinations it requires is much more strictly defined than in the Method of Agreement. The two instances that are to be compared must be exactly similar in all circumstances except the one we’re trying to investigate; they must inter-relate as A B C relates to B C (·if we’re investigating the effects of A·) or as a b c relates to b c (·if we’re investigating the cause of a·). This similarity of circumstances needn’t be total—it needn’t extend to circumstances that we already know to be irrelevant to the result. With most phenomena we learn at once, from the commonest experience, that most of the coexistent phenomena in the universe can be either present or absent without affecting the given phenomenon. . . . Still, even limiting the identity that’s required between the two instances A B C and B C to circumstances that aren’t already known to be irrelevant, nature very seldom offers two instances that we can be sure are related in that way—·i.e. that the only difference between them (apart from ones that we know are irrelevant) is the presence of A in one of them and not the other·. Nature’s spontaneous operations are generally so complicated and so obscure—being out of our reach because they are too vast or too tiny—that we’re ignorant of a great part of the facts that really take place, and even the ones we aren’t ignorant of are so numerous and thus so seldom exactly alike in any two cases that a spontaneous experiment [= ‘a hands-off observation’] of the kind required by the Method of Difference is usually not to be found. On the other hand, when we obtain a phenomenon by an artificial experiment, a pair of instances such as the method requires is obtained almost as a matter of course, provided the process doesn’t last a long time. A certain state of surrounding circumstances existed before we started the experiment; this is B C. We then introduce A—e.g. by merely bringing an object from another part of the room—before there has been time for any change in the other elements. Comte was right: it’s the very nature of an experiment to introduce into the pre-existing state of circumstances a perfectly definite change. We choose a previous state of things that we are well acquainted with, so that it’s not likely to change without our noticing; and into this we introduce, as rapidly as possible, the phenomenon x that we want to study; so that in general we’re entitled to be quite sure that the state we have produced differs from the pre-existing state only in the presence or the absence of x. If a bird is taken from a cage and instantly plunged into carbonic acid gas, the experimenter can be fully assured (after one or two repetitions) that no circumstance that could cause suffocation had intruded except the change from •immersion in the atmosphere to •immersion in carbonic acid gas. . . . It thus appears that in the study of the various kinds of phenomena that we can modify or control, we can in general satisfy the requirements of the Method of Difference; but that those requirements are seldom fulfilled by the spontaneous operations of nature.

With the Method of Agreement the situation is reversed. We don’t here require instances of such a special and determinate kind. For the purposes of this method, any instances in which nature presents us with a phenomenon can be examined, and if all such instances agree in anything, that’s already a useful conclusion. It’s true that we usually can’t be sure that the one point of agreement is the only one; but this ignorance does not invalidate the conclusion, as it would with the Method of Difference. . . . We have ascertained one invariable antecedent or consequent, however many other invariable antecedents or consequents may still remain unascertained. If A B C and A D E and A F G are all equally followed by a then a is an invariable consequent of A. If a b c and a d e and a f g all have A among their antecedents, then A is connected as an antecedent with a. But to determine whether this invariable antecedent is a cause, or this invariable consequent an effect, we must also be able to produce one of them by means of the other; or at least to obtain an instance in which the effect a has come into existence with no change in the circumstances except the addition of A. (That is our only way of being sure that we have produced something.) And this, if we can do it, is an application of the Method of Difference, not of the Method of Agreement.

So it seems that the only way direct experience can give us certain results about causes is through the Method of Difference. The Method of Agreement leads only to uniformities which either aren’t laws of causation or whose status as causal must for the present remain undecided. (Some writers call these ‘laws of phenomena’, but that’s a bad usage because laws of causation are also laws of phenomena.)

The Method of Agreement is to be used mainly •as a means of suggesting applications of the Method of Difference (as in the last example, where the comparison of A B C and A D E and A F G suggested that A was the antecedent on which to try by experiment whether it could produce a); or •as a second-best in cases where the Method of Difference is impracticable—e.g. because we can’t artificially produce the phenomena. So the Method of Agreement—though applicable in theory to either case—is more emphatically the method of investigation in cases where artificial experimentation is impossible, because in them it’s usually our only resource of a directly inductive kind, whereas with phenomena that we can produce at will the Method of Difference is generally more effective because it can ascertain •causes as well as •mere laws.

§4. But in many cases our power of producing the phenomenon is complete and yet the Method of Difference can’t be used at all, or only with a previous use of the Method of Agreement. This occurs when our only way of producing the phenomenon involves a combination of antecedents that we can’t separate from each other and exhibit apart. Suppose, for instance, that we want to investigate the cause of the double refraction of light. We can produce this phenomenon at will, using any one of the many substances that we know to refract light in that special manner—Iceland spar, for example—but we can’t use the Method of Difference because we can’t find another substance precisely resembling Iceland spar except in some one property. The only way to push this inquiry is the one provided by the Method of Agreement. And that’s what was used: the physicists compared all the known substances that doubly refract light, and found that they have in common being crystalline substances; from which they reasonably inferred. . . . that either •crystalline structure or •the cause of that structure is one of the conditions of double refraction.

[This paragraph will have a good many small omissions not indicated by. . . .ellipses. The reasons are purely aesthetic; you can trust the paragraph’s content.] Suppose that by using the Method of Agreement we have discovered that there’s a connection between A and a. To convert this evidence of connection into proof of causation by the direct Method of Difference we would need to do things like:

having tested A B C and found that it leads to a, we then test B C and observe whether that leads to a also.

Now, we often can’t do this (see the Iceland spar example), but sometimes we can find out what would be the upshot if we could test B C, and that’s as good as conducting the test. Here’s how we do that:

Having tested a variety of cases where a occurred, and found that they all contain A, we now observe a variety of instances where a doesn’t occur, and find that none of them contains A.

This establishes by the Method of Agreement the same connection between the absence of A and the absence of a, which was previously established between their presence. Just as our first work showed that whenever A is present a is present, so now we can conclude whenever A is absent a is also absent, which means that we have the positive and negative instances that the Method of Difference requires.

This method—call it the ‘Indirect Method of Difference’ or the ‘Joint Method of Agreement and Difference’—consists in two uses of the Method of Agreement, each independent of the other and corroborating it. But it isn’t equivalent to a proof by the direct Method of Difference. The Method of Difference requires us to be quite sure •that the instances leading to a have nothing in common except A, or •that the instances that don’t lead to a have nothing in common but the absence of A. This is never possible; and if it were, we wouldn’t need the joint method, because either of the two sets of instances separately would prove causation. This indirect method, therefore, can only be regarded as an extension and improvement of the Method of Agreement, but not as having any part in the more powerful nature of the Method of Difference. Its canon is this:

THIRD CANON.

If two or more instances in which the phenomenon occurs have only one circumstance x in common, while two or more instances in which it doesn’t occur have nothing in common except the absence of x, then x is the effect, or the cause or an indispensable part of the cause, of the phenomenon.

[Mill says that the Joint Method of Agreement and Difference has another advantage over ‘the common Method of Agreement’, but that he needs to postpone discussing this until later, and will] at once proceed to a statement of the other two methods, which will complete the list of the means we have for exploring the laws of nature by specific observation and experience.

§5. The first of these has been well named ‘the Method of Residues’. Its principle is very simple. Remove from any given phenomenon all the parts of it that can by virtue of preceding inductions be assigned to known causes, and what’s left will be the effect of antecedents which had been overlooked or whose effect was still an unknown quantity.

Suppose again that we have the antecedents A B C followed by the consequents a b c, and that by previous inductions (based, let’s say, on the Method of Difference) we have discovered the causes of some of these effects or the effects of some of these causes; specifically we have learned that the effect of A is a, and that the effect of B is b. Subtracting the sum of these effects from the total phenomenon, there remains c, and we don’t need any new experiments to know that c is the effect of C. This Method of Residues is in fact a special adaptation of the Method of Difference. If the instance A B C and a b c could have been compared with a single instance A B and a b, we would have proved C to be the cause of c by using the Method of Difference in the ordinary way. In the present case, though, instead of a single instance A B we have had to study separately the causes A and B, and to infer from the effects they produce separately what effect they must produce in the case A B C, where they act together. Thus, of the two instances that the Method of Difference requires—one positive, the other negative—the negative one (in which the given phenomenon is absent) is not the direct result of observation and experiment, but has been arrived at by deduction. As one of the forms of the Method of Difference, the Method of Residues shares in its rigorous certainty, provided the previous inductions—the ones that gave the effects of A and B—were obtained by the same infallible method, and provided we’re certain that C is the only antecedent that the residual phenomenon c can be connected with, i.e. the only agent whose effect we hadn’t already calculated and subtracted. But we can never be quite certain of this, so the evidence derived from the Method of Residues is not complete unless we can obtain C artificially and test it separately, or unless its agency, when once suggested, can be explained and proved deductively from known laws.

Even with these reservations, the Method of Residues is one of our most important instruments of discovery. Of all the methods of investigating laws of nature, this is the most fertile in unexpected results, often informing us of sequences in which neither the cause nor the effect was conspicuous enough to attract the attention of observers. The agent C may be an obscure circumstance, not likely to have been perceived unless sought for, nor likely to have been sought for until attention had been awakened by the insufficiency of the obvious causes to account for the whole of the effect. And c may be so disguised by its intermixture with a and b that it would scarcely have presented itself spontaneously as a subject of separate study. I’ll soon present some remarkable examples of these uses of the Method of Residues. Its canon is as follows:

FOURTH CANON.

Subtract from any phenomenon the part of it that is known by previous inductions to be the effect of certain antecedents, and the remainder of the phenomenon is the effect of the remaining antecedents.

§6. There remains a class of laws that can’t be discovered by any of the three methods I have tried to describe, namely the laws of •permanent causes—i.e. •indestructible natural agents—that we can’t exclude or isolate, can’t hinder from being present or arrange to have present alone. You might think that we can’t possibly separate the effects of these agents from the effects of the other agents that they have to coexist with; but in fact for most of the permanent causes no such difficulty arises: although we can’t eliminate them as •coexisting facts, we can eliminate them as •influencing agents by simply conducting our experiment in a place outside the reach of their influence. The swing of a pendulum, for example, is disturbed by a nearby mountain; we move the pendulum far enough away from the mountain, and the disturbance ceases. From these data [see Glossary] we can use the Method of Difference to calculate the amount of effect due to the mountain; and beyond a certain distance everything goes on precisely as it would do if the mountain exercised no influence whatever, and we reasonably enough conclude that it doesn’t.

But the picture changes when we can’t get ourselves ·or our experimental apparatus· out of reach of the influence of a permanent cause. The pendulum can be moved away from the influence of •the mountain, but it can’t be removed from the influence of •the earth; we can’t move the earth and the pendulum away from one another, to discover whether it would continue to swing if the earth’s action on it were withdrawn. Then what is our evidence that the pendulum’s swing is caused by the earth’s influence? It can’t be anything supported by the Method of Difference, for one of the two instances is lacking—namely the negative instance where the earth’s influence isn’t a factor. Nor by the Method of Agreement: when any pendulum swings the earth is always present, but so is the sun! Obviously to establish even such a simple fact of causation as this we needed some method other than those I have so far presented.

For another example, consider heat. Independently of any theory about the real nature of heat we can be sure of this much: •we can’t deprive any body of the whole of its heat, and •no-one ever perceived heat that wasn’t being given off by a body. So we can’t separate body and heat, and therefore can’t vary the circumstances in the way the foregoing three methods require—we can’t ascertain by those methods what portion of the phenomena exhibited by any body is due to the heat contained in it. If we could observe a body with its heat, and the same body entirely divested of heat, the Method of Difference would show the effect of the heat, apart from the effect of the body. If we could observe heat under circumstances agreeing only in heat, and therefore not involving the presence of a body, we could use the Method of Agreement to discover the effects of heat by comparing •an instance of heat with a body and •an instance of heat without a body; or we could use the Method of Difference to discover what effect was due to the body, the remainder due to heat being given by the Method of Residues. But we can’t do any of these things, so none of the three methods can help us to solve this problem. . . .

. . . .But there is still something we can try. Even when we can’t exclude an antecedent altogether, we may be able to produce—or nature may produce for us—some modification in it, by which I mean a change in it not amounting to its total removal. If some modification in the antecedent A is always followed by a change in the consequent a, the other consequents b and c remaining the same; or vice versa if every change in a is found to have been preceded by some modification in A, none being observable in any of the other antecedents, we can safely conclude that a is at least in part causally connected somehow with A. We can’t expel heat altogether from any body, but we can modify its amount, increasing or diminishing it; and in doing this we can find by the various methods of experiment or observation that I have discussed that such increase or diminution of heat is followed by expansion or contraction of the body. This brings us to the conclusion that we couldn’t have achieved in any other way, that one effect of heat is to make bodies bigger, i.e. to increase the distances between their particles.

A change in a thing that doesn’t amount to its total removal—i.e. a change that leaves it still the same thing— must be a change either in •its quantity or in •some of its variable relations to other things; and the main one of these is position in space. We have seen an example depending on quantity; now for one involving spatial position. Question: what influence does the moon exert on the surface of the earth? We can’t try an experiment in the absence of the moon. But when we find that all the variations in the moon’s position are followed by corresponding variations in the time and place of high tide, the place always being either the part of the earth nearest to the moon or the part furthest from it, this gives us ample evidence that the moon is at least partially the cause that determines the tides. . . .

Similar evidence shows that the swinging of a pendulum is caused by the earth. The swings take place between equidistant points on opposite sides of a line that •is perpendicular to the earth, and therefore •varies with every variation in the earth’s position. . . . This method tells us that all terrestrial bodies tend toward the earth, and not towards some unknown fixed point lying in the same direction. In every 24 hours of the earth’s rotation, the line drawn from the body at right angles to the earth coincides successively with all the radii of a circle, and in the course of six months the place of that circle changes by nearly 200,000,000 million miles; yet in all these changes of the earth’s position the line in which bodies tend to fall—·the line down the centre of the pendulum’s swing·—continues to be directed toward it. This proves that terrestrial gravity is directed towards the earth and not, as some people used to think, towards a fixed point in space.

The method by which these results were obtained may be termed the Method of Concomitant [see Glossary] Variations; it is regulated by the following canon:

FIFTH CANON.

If any phenomenon x varies in some specific way whenever another phenomenon y varies in some specific way, x is either a cause or an effect of y, or is causally connected with it in some other manner.

I add that last clause because when two phenomena match each other in their variations it doesn’t follow one is cause and the other effect. If they were two effects of a common cause, they would exhibit concomitant variation; and this method alone can’t tell us whether they’re related in that way rather than as cause and effect. The only way to answer the question would be—yet again!—by trying to ascertain whether we can produce the one set of variations by means of the other. In the case of heat, for example, by increasing the temperature of a body we make it bigger, but by making it bigger (e.g. by using an air-pump to decrease the air-pressure on it) we don’t increase its temperature; on the contrary, in most cases we diminish it. So heat is not an effect of increase in size but a cause of it. If we can’t ourselves produce the variations, we must try—though we’ll usually fail—to find them produced by nature in some case in which the pre-existing circumstances are perfectly known to us. . . .

You might think that the Method of Concomitant Variations assumes a new axiom, i.e. a new law of causation in general, namely: Every modification of the cause is followed by a change in the effect. And it does usually happen that when a phenomenon A causes a phenomenon a, any variation in A’s quantity or relational properties is uniformly followed by a variation in the quantity or relational properties of a. . . . The sun causes a certain tendency to motion in the earth; here we have cause and effect; but that tendency is toward the sun, and therefore varies in •direction as the sun varies in its position relative to the earth; and the tendency also varies in •intensity in a certain numerical correspondence to the sun’s distance from the earth—i.e. according to another relation of the sun. So there’s not only an invariable connection between the sun and the earth’s gravitation, but two of the sun’s relational properties—its position relative to the earth and its distance from the earth—are invariably connected as antecedents with the quantity and direction of the earth’s gravitation. The cause of the earth’s gravitating at all is simply the sun; but the cause of its gravitating with a given intensity in a given direction is the existence of the sun at a given distance and in a given direction from the earth. A modified cause is really a different cause, so it’s not surprising that it produces a different effect.

But the Method of Concomitant Variations doesn’t require as an axiom that

(a) If x is the cause of y, any modification of x is followed by a modification of y.

All it needs is the converse proposition:

(b) If every modification of x is followed by a modification of y, x is the cause of y (or is connected with the cause of y).

It’s obvious that (b) is true, because if x has no influence over y then modifications of x can’t influence y either. If the stars have no power over the fortunes of mankind, then the conjunctions or oppositions of stars can have no such power.

The most striking uses of the Method of Concomitant Variations occur in cases where the Method of Difference, strictly so-called, is impossible; but its use isn’t confined to those cases. It is often useful as a follow-up to the Method of Difference, to give additional precision to a solution that the latter method has found. When we know through the Method of Difference that x produces y, the Method of Concomitant Variations can be usefully called in to determine what law governs the match between x’s quantity and relational properties and y’s.

§7. This method is most widely used in cases where the ·relevant· variations of the cause are variations of quantity. It’s pretty safe to say that quantitative variations in the cause will be attended by quantitative variations in the effect; because the proposition that •more of the cause is followed by •more of the effect follows from the principle of the Composition of Causes, which we saw here to be the •general rule of causation, whereas counterexamples to it—cases where causes change their properties on being combined—are •special and exceptional. Suppose that when A changes in quantity, a also changes in quantity, and that we can trace the •numerical relation between parts of the two sets of changes—the parts, that is, that aren’t too big or too small for us to observe them. Then with certain precautions we can safely conclude that the same •numerical relation will hold outside those limits. If we find that when A is double, a is double, when A is treble or quadruple, a is treble or quadruple, we can conclude

(i) that if A were a half or a third, a would be a half or a third, and

(ii) that if A were annihilated, a would be annihilated; and thus

(iii) that a is wholly the effect of A or wholly the effect of A’s cause.

And we could infer (iii) for any other numerical relation according to which (ii) A and a would vanish simultaneously— e.g. if a were proportional to A^2. If on the other hand a is not wholly the effect of A, but still varies when A varies, it is probably a mathematical function not of A alone but of A and something else. For example, its changes may be what you would get if some part of it remained constant or varied on some other principle, while the remainder varied in some numerical relation to the variations of A. In that case, as A diminishes, a will be seen to approach not •zero but •some other limit; and when the series of variations indicates what that limit is, the limit will exactly measure how much of a is the effect of some other and independent cause, and the remainder will be the effect of A (or of the cause of A). That is stated for cases where the limit is constant; if it is variable, replace ‘indicates what that limit is’ by ‘indicates what the law of its variation is’.

But these conclusions mustn’t be drawn without certain precautions. In the first place, they can’t be drawn at all unless we’re acquainted not only with •the variations but with •the absolute quantities both of A and a. If we don’t know the total quantities, we can’t determine the numerical relation according to which they vary. So it’s an error to conclude (as some have concluded) that because increase of heat expands bodies, i.e. increases the distance between their particles, therefore •that distance is wholly the effect of heat, and •if we could entirely deprive the body of its heat the particles would be in complete contact. This is a mere guess, and wildly risky one rather than a legitimate induction. Because we don’t know how much heat there is in any body, or what the real distance is between any two of its particles, we can’t judge whether the contraction of the distance follows the diminution of the quantity of heat according to a numerical relation such that the two quantities would reach zero simultaneously.

Now consider a case where the absolute quantities are known, namely the case addressed in the first law of motion, which says that all bodies in motion continue to move in a straight line with uniform velocity until acted upon by some new force. This is in open opposition to first appearances; all moving terrestrial objects slow down and eventually stop; and the ancients—going by inductio per enumerationem simplicem [see Glossary]—imagined that to be the law. But every moving body encounters various obstacles—friction, the resistance of the atmosphere, etc.—which we know by daily experience to be causes that can destroy motion. It was suggested that the lessening of motion might come wholly from these causes. How was this inquired into? With the obstacles entirely removed, the Method of Difference could have come into play. But they couldn’t be removed, only lessened, so the case had to be handled by the Method of Concomitant Variations. This was used, and it was found that every lessening of the obstacles lessened the slowing of the motion; and this being a case (unlike the case of heat) where the total quantities of both the antecedent and of the consequent were known, it was possible to get a fairly accurate estimate of the amount of •the slowing and the amount of •the relevant resistances, and to judge how near each was to zero; and it turned out that the effect dwindled as rapidly as the cause did, so that at each step the two were equally near to annihilation. The swinging of a weight suspended from a fixed point and moved a little out of the perpendicular ordinarily lasts for only a few minutes, but Borda got it to continue for more than thirty hours by going as far as possible towards reducing the friction at the point of suspension and making the body move in a vacuum. That left no reason to hesitate to conclude that the whole of the slowing of motion was due to the influence of the obstacles. With the slowing removed from the total phenomenon, the remainder was a uniform velocity, and the result was the proposition known as the first law of motion.

The inference that the law of variation that the quantities conform to within our limits of observation will hold beyond those limits is open to another kind of uncertainty. Actually there are two of them, one being obvious: we don’t know what happens in the range outside the limits of our observation, and it might be that something comes into play there that spoils our conclusion. This kind of uncertainty comes into virtually all our predictions of effects; it’s not specially relevant to the Method of Concomitant Variations. I want to talk about an uncertainty that is characteristic of that method; especially in the cases where our observable range is very small compared with the possible variations in the quantities of the phenomena. If you know anything of mathematics you know that very different laws of variation can produce numerical results that differ only slightly from one another; and in many cases it’s only when the absolute amounts of variation are considerable that we can see the difference between the results given by two rival laws. The upshot is that when the variations in the quantity of the antecedents that we can observe are small in comparison with the total quantities, there’s a great danger of our picking the wrong numerical law, and being led to miscalculate the variations that would occur beyond our limits. That miscalculation would invalidate any conclusion about the dependence of the effect on the cause. There are plenty of examples of such mistakes. Herschel writes: ‘The formulae that have been empirically deduced for the elasticity of steam (till very recently), and those for the resistance of fluids and other similar subjects’, when relied on beyond the limits of the observations from which they were deduced, ‘have almost invariably failed to support the theoretical structures based on them’.

Even when we have this uncertainty, the Method of Concomitant Variation can prove that there is some connection between A and a, and. . . .can legitimately satisfy us that the relation we have observed (within our limits) to exist between the variations of A and a will hold true in all cases that fall between those same limits. . . .

The four methods that I have tried to describe are the only possible modes of experimental inquiry—of direct induction a posteriori as distinguished from deduction. At any rate, I don’t know of any others and can’t imagine any others. And the Method of Residues (I remind you) isn’t independent of deduction; but I include it among methods of direct observation and experiment because as well as deduction it also requires specific experience.

. . . .In chapter 10 I’ll come to certain circumstances that make the use of these methods much more complicated and difficult ·than I have so far indicated·. Before coming to that, though, I shall illustrate the use of the methods by suitable examples drawn from actual physical investigations.

Chapter 9. Examples of the four methods

§1. First example: I’ll start with an interesting bit of theory by one of the most eminent theoretical chemists, Baron Liebig. The objective is to discover the immediate cause of the death produced by metallic poisons.

Arsenious acid and the salts of lead, bismuth, copper, and mercury, if introduced into the animal organism in anything but the smallest doses, destroy life. These facts have long been known, as separate and unconnected truths that are as ungeneral as generalisations can be. It was left to Liebig, by an apt employment of the Methods of Agreement and Difference, to connect these truths with one another by a higher induction, revealing the property that •is common to all these harmful substances and •is the operative cause of their fatal effect.

(a) When solutions of these substances are placed in close enough contact with many animal products—albumen, milk, muscular fibre, and animal membranes—the acid or salt leaves the water it was dissolved in and enters into combination with the animal substance; and this substance, after being thus acted upon, is found to have lost its tendency to putrefy [see Glossary].

(b) Observation also shows, in cases where death has been produced by these poisons, that the parts of the body that the poisonous substances have been brought into contact with don’t afterwards putrefy.

(c) And, finally, when the poison has been supplied in too small a quantity to destroy life. . . ., certain superficial portions of the tissues are destroyed and afterwards thrown off by the process of recovery in the healthy parts.

These three sets of instances can be handled according to the Method of Agreement. In all of them the metallic compounds are brought into contact with the substances that compose the human or animal body; and the instances seem to have nothing else in common. The remaining antecedents are as different—even opposite—as they could possibly be made; for in some the animal substances exposed to the action of the poisons are in a state of life, in others only in a state of organisation, in others not even in that. And the result in all the cases is the conversion of the animal substance (by combination with the poison) into a chemical compound that is held together by force so powerful that it resists the subsequent action of the ordinary causes of decomposition. Now, organic life (the necessary condition of sensitive life) consists in a continual state of decomposition and recomposition of the different organs and tissues, so anything that prevents this decomposition destroys life. Thus the immediate cause of the death produced by poisons of this kind is ascertained, as far as the Method of Agreement can ascertain it.

Let us now use the Method of Difference to test our conclusion. This will involve a comparison. On one hand we have:

cases where the antecedent is the presence of substances that combine with the tissues to form a compound that can’t putrefy (and therefore can’t support life), the consequent being death of the whole organism or of some part of it.

We are to compare these with

cases as much like the former ones as possible except that they don’t have the death of anything as their effect.

Many insoluble basic salts of arsenious acid are known not to be poisonous. The substance called ‘alkargen’, discovered by Bunsen, which contains a great amount of arsenic and is very like the organic arsenious compounds found in the body, hasn’t the slightest injurious action upon the organism. Now when these substances are brought into contact with the tissues in any way, they don’t combine with them, and don’t stop their progress towards decomposition. What these instances seem to show is that when the effect is absent that’s because of the absence of the antecedent that we already had good reason to consider as the immediate cause.

But the rigorous conditions of the Method of Difference aren’t yet satisfied; for we can’t be sure that these unpoisonous bodies differ from the poisonous substances only in not combining with animal tissues to form a compound that resists decomposition. To make the method strictly applicable, we need an instance not of a different substance but of one of the very same substances, in circumstances that prevent it from combining with the tissues to form the sort of compound in question; and then, if death doesn’t follow, our case is made out. Instances of this kind are provided by the antidotes to these poisons. For example, if hydrated peroxide of iron is administered along with poisonous arsenious acid, the destructive agency of the latter is instantly checked. Now, this peroxide is known to combine with the acid to form a compound that is insoluble, and so can’t act at all on animal tissues. Thus, sugar is a well-known antidote to poisoning by salts of copper; and sugar turns those salts into something that doesn’t combine with animal matter. The disease called ‘painter’s colic’, so common in factories making white lead, is unknown where the workmen regularly take (as a preservative) a solution of sugar made acid by sulphuric acid. Now, diluted sulphuric acid has the property of •decomposing all compounds of lead with organic matter or •preventing them from being formed.

[Mill then describes a set of facts about ‘soluble salts of silver’ which, when applied externally, have about the same effect as arsenious acid, but aren’t poisonous when ingested. The explanation is that the animal stomach contains common salt and muriatic acid, which turn the soluble salts into something virtually insoluble and therefore unable to combine with the tissues to fatal effect,]

Those instances have shown us a very conclusive induction that illustrates the two simplest of our four methods; though it doesn’t rise to the maximum of certainty that a perfect example of the Method of Difference can provide. Remember that the positive instance and the negative one strictly ought to differ only in the presence or absence of one single circumstance. And in the foregoing argument they differ in the presence or absence not of a single •circumstance but of a single •substance; every substance has countless properties; so there’s no knowing how many real differences are involved in what is apparently only one difference. It is conceivable that the antidote. . . .counteracts •the poison through some property other than that of forming an insoluble compound with •it; and if that were so the theory would collapse so far as it rests on that instance. This source of uncertainty is a serious hindrance to all extensive generalisations in chemistry; but in our present case it is reduced to almost the lowest possible degree when we find that many substances can act as antidotes to metallic poisons, and that all these share the property of forming insoluble compounds with the poisons and can’t be ascertained to share any other property whatsoever. So we have in favour of the theory all the evidence that can be obtained by the Joint Method of Agreement and Difference [see here]; and though the evidence it produces can’t amount to that of the Method of Difference properly so-called, it can approach indefinitely near to that.

§2. Second example: The aim is to discover the law governing ‘induced electricity’—i.e. to learn under what conditions a body that is positively or negatively electrified gives rise to the opposite electric state in some other body adjacent to it.[1]

The most familiar kind of example of the phenomenon to be investigated is the following. Around the prime conductors of an electrical machine the nearby atmosphere or any conducting surface suspended in it is found to be in the electric condition opposite to that of the prime conductor itself: near and around the positive prime conductor there’s negative electricity, and near and around the negative prime conductor there’s positive electricity. When a pith ball (or a human hand) is brought near to one of the conductors, it becomes electrified with the opposite electricity to it—either •receiving a share from the already electrified atmosphere by conduction, or •acted on by the direct inductive influence of the conductor itself—and then it is attracted by the conductor to which it is opposite or by any other oppositely charged body. Now, we have no evidence that a charged conductor can be suddenly discharged except by the approach of a body with the opposite charge. In the case of the electric machine, therefore, it appears that the accumulation of electricity in an insulated conductor is always accompanied by the excitement of the opposite electricity in •the surrounding atmosphere and in things in •it. It does not seem possible, in this case, to produce one electricity by itself. [That last sentence is verbatim from Mill.]

Let us now examine all the other instances we can get that resemble this one in the given consequent, namely the occurrence of an opposite electricity in the neighbourhood of an electrified body. One remarkable instance is the Leyden jar; another is the magnet, in which it is impossible to produce one kind of electricity by itself, i.e. to charge one pole without charging another pole with the opposite electricity at the same time. (That holds both for natural magnets and for electromagnets. In counting magnets as relevant to my topic, I am relying on Faraday’s splendid experiments decisively showing that magnetism and electricity are basically the same thing.) We can’t have a magnet with one pole; if we break a natural lodestone into a thousand pieces, each piece will have its two oppositely electrified poles complete within itself. In the voltaic circuit, again, we can’t have one current without its opposite. In the ordinary electric machine, the glass cylinder or plate acquires an electrical charge opposite to that of the rubber.

From all these instances, treated by the Method of Agreement, a general law appears to result. The instances cover all the known ways in which a body can get an electric charge; and in all of them there is found, as a concomitant or consequent, the excitement of the opposite electric charge in some other body or bodies. It seems to follow that the two facts are invariably connected, and that a necessary condition of a body’s acquiring an electric charge is the simultaneous excitement of the opposite charge in some neighbouring body.

As the two opposite charges can only be produced together, so they can only cease together. This can be shown by an application of the Method of Difference to the Leyden jar. In the Leyden jar electricity can be accumulated and retained in considerable quantity, by the device of having two conducting surfaces of equal extent, and parallel to each other through the whole of that extent, with a non-conducting substance such as glass between them. When one side of the jar is charged positively, the other is charged negatively (which is why I cited the Leyden jar as an instance in our use of the Method of Agreement). Now, it’s impossible to discharge one of the coatings unless the other is discharged at the same time. A conductor held to the positive side can’t convey away any electricity unless an equal quantity is allowed to pass from the negative side; if one coating is perfectly insulated, the charge is safe. . . .

The law that this strongly indicates can be corroborated by the Method of Concomitant Variations. The Leyden jar can receive a much higher charge than can ordinarily be given to the conductor of an electrical machine. Now, in the Leyden jar the metallic surface that receives •the induced electricity is a conductor exactly like that which receives •the primary charge, and is therefore as capable of receiving and retaining one charge as the opposite surface is of receiving and retaining the other; but in the machine the neighbouring body that is to get the opposite charge is the surrounding atmosphere or a nearby object; and as these can usually hold only a much smaller charge than the conductor itself, their limited power imposes a corresponding limit to the conductor’s capacity for being charged. As the neighbouring body’s ability to support the opposition increases, a higher charge becomes possible; and that appears to explain the great superiority of the Leyden jar.

One of Faraday’s experiments provides a further and most decisive confirmation by the Method of Difference. [Mill’s account of the experiment and the conclusion drawn from it is hard to follow. It speaks of ‘two opposite ·electric· currents. . . .both accommodated in one wire’, and it’s hard to see what Mill has in mind. We can slide past this example without harm to our grasp of the rest of what he has to say.]

§3. Our third example will be extracted from Herschel’s Discourse on the Study of Natural Philosophy, a work full of well-selected examples of inductive processes from almost every branch of physical science. . . . The present example is described by Herschel as ‘one of the most beautiful specimens’ that can be cited ‘of inductive experimental inquiry lying within a moderate compass’—namely, the theory of dew that is now accepted by all scientific authorities.

[Mill devotes four pages to this, much of it in direct quotations from Herschel. We can afford to excuse ourselves from going through all the details. Mill shows that the series of tests and experiments make clear use of three of his methods (the exception being the Method of Residues). At a certain point he arrives at this:]

It thus appears that the various instances in which much dew is deposited agree in this (and as far as we can see only this): they either •radiate heat rapidly or •conduct it slowly; and those two qualities have nothing in common except that by virtue of either of them the body tends to lose heat from the surface faster than it can be restored from within. And the instances where little or no dew is formed have nothing in common (as far as we can see) except not having this same property. So we seem to have detected the characteristic difference between the substances on which dew is produced and those on which it isn’t produced. We have done this by using the Joint Method of Agreement and Difference; and the data were prepared for that by the Methods of Agreement and of Concomitant Variations. . . .

Can we be quite sure that the substances on which dew is produced differ from those on which it isn’t in nothing but the property—·I’ll call it R·—of losing heat from the surface faster than the loss can be repaired from within? No, but this matters less than you might think. Suppose there is an undiscovered property Q that is present in all the substances that contract dew and absent from those that don’t, Q must be present in all the substances that have R and in none of the substances that don’t. That much match between two properties creates a strong presumption that they have the same cause and therefore will invariably go together. And if that is right, then the property R—being a better radiator than conductor—if it isn’t itself the cause almost certainly always accompanies the cause, and for purposes of prediction we can safely treating it as if it really were such.

At an earlier stage of the inquiry we found that whenever dew is formed the surface on which it forms is colder than the surrounding air. Was this coldness the cause of dew or an effect of it? We can now answer this. We have found that when dew forms, the substance on which it forms is one which, by its own properties or laws, would if exposed in the night become colder than the surrounding air. •The coldness is accounted for independently of the dew, while it is proved that •there is a connection between the two; so it must be the case that •the dew depends on the coldness, i.e. that the coldness is the cause of the dew.

This law of causation, already so amply established, can be further corroborated in no less than three ways. (i) First, by the Deductive Method. I won’t be ready to deal with that until chapter 11, but I’ll say enough here to firm up the results concerning dew. It is known by direct experiment that only a limited quantity of water can remain suspended as vapour at each degree of temperature, and that this maximum goes down as the temperature falls. From this it follows deductively that if the air already has much vapour as it can contain at its existing temperature, any lowering of that temperature will cause a portion of the vapour to be condensed. And we also know deductively, from the laws of heat, that the air’s contact with a body colder than itself must lower the temperature of the layer of air immediately against its surface, and will therefore cause it to part with some of its water. And this, by the ordinary laws of gravitation or cohesion—·deduction again!·—will attach itself to the surface of the body, constituting dew. This deductive proof has the advantage of proving causation as well as coexistence; and it has the further advantage of explaining the exceptions, the cases where the body is colder than the air but no dew is deposited—by showing that this must be the case when the air has too little vapour to give any of it up. That’s why in a very dry summer there are no dews, and in a very dry winter no hoar-frost. This is a condition of the production of dew that wasn’t detected by the other methods; it might have remained still undetected if we hadn’t set out to deduce the effect from the known properties of the agents known to be present.

(ii) The second corroboration is by direct experiment according to the canon of the Method of Difference. By cooling the surface of a body we can find the temperature at which dew begins to be deposited. Here again the causation is directly proved. We can accomplish this only on a small scale, but we have ample reason to conclude that the same operation, if conducted in •nature’s great laboratory, would equally produce the effect.

(iii) Even on •that great scale we can verify the result. This is one of the rare [see here] cases where nature works the experiment for us in the same way that we ourselves perform it, introducing into the previous state of things a single perfectly definite new circumstance, and producing the effect so rapidly that there’s no time for any other material [see Glossary] change in the pre-existing circumstances. Herschel writes:

‘It is observed that dew is never copiously deposited in situations much screened from the open sky, and not at all in a cloudy night; but if the clouds withdraw even for a few minutes and leave a clear opening, dew starts to appears almost at once, and goes on increasing. . . Dew formed in clear intervals often evaporates when the sky becomes thickly overcast.’

So we have complete proof that the presence or absence of an uninterrupted communication with the sky causes the deposition or non-deposition of dew. Now, a clear sky is merely the absence of clouds, and we know that clouds. . . .tend to raise or keep up the surface temperature of a nearby object by radiating heat to it; so we see at once that the disappearance of clouds will cause the surface to cool. Thus, in this case nature produces a change in the antecedent by definite and known means, and the consequent follows accordingly—a natural experiment that satisfies the requirements of the Method of Difference![2]

The accumulated proof that has been found for the theory of dew is a striking example of the fullness of assurance that the inductive evidence of laws of causation can achieve in cases where the invariable sequence is far from obvious at first glance.

§4. Fourth example: The admirable physiological investigations of Brown-Séquard provide brilliant examples of the use of the inductive methods in a class of inquiries in which—for reasons I’ll give soon—direct induction is done under special difficulties and disadvantages. I select his theorising. . . .about the relations between •muscular irritability [see Glossary], •rigor mortis [see Glossary], and •putrefaction.

The law that Brown-Séquard’s investigation tends to establish, is this:

The greater the degree of muscular irritability at the time of death, the later the rigor mortis sets in, and the longer it lasts, and also the later putrefaction appears, and the more slowly it progresses.’

At first glance you’d think that this must be work for the Method of Concomitant Variations, but that is wrong—it’s an illusion arising from the fact that the conclusion to be tested is itself a fact about concomitant variations. For the establishment of that fact any of the ·four· Methods may be put to work, and it will turn out that the fourth Method—·the Method of Concomitant variations·—has a real but subordinate place in this investigation.

The items of evidence by which Brown-Séquard establishes the law can be enumerated as follows:

Firstly: (a) •Paralysed muscles have greater irritability than healthy muscles. And (b) paralysed muscles are later in entering rigor mortis than healthy muscles, the rigor lasts longer, and putrefaction sets in later and proceeds more slowly.

Brown-Séquard proved both these propositions by experiment. He established (a) in various ways, but most decisively by comparing the duration of irritability in a paralysed muscle and in the corresponding healthy muscle on the opposite side when they are both submitted to the same stimulus. He often found that the paralysed muscle remained irritable up to four times as long as the healthy one. This is induction by the Method of Difference. Because the two limbs were those of the same animal, they were presumed not to differ in any circumstance relevant to the case except the paralysis, so that the presence and absence of paralysis was the source of the difference in the muscular irritability. The assumption that there was only one relevant difference between the legs wasn’t safe in any one pair of experiments, because the two legs of any given animal might happen to differ in other relevant respects; but if. . . .the experiment was repeated often enough with different animals to exclude the supposition that any abnormal circumstance could be present in them all, the conditions of the Method of Difference were well enough satisfied.

Brown-Séquard also proved the proposition (b) concerning rigor mortis and putrefaction. Having. . . .cut some nerves so as to produce paralysis in one hind leg of an animal but not the other, he found that muscular irritability lasted much longer in the paralysed limb, rigor set in later and ended later, and putrefaction began later and progressed more slowly than on the healthy side. This is a routine use of the Method of Difference, requiring no comment. An important corroboration was obtained by the same method. When the animal was killed not •soon after the nerves were cut but •a month later, the effect was reversed; rigor set in sooner and lasted a shorter time in the paralysed limb than in the healthy one. What had happened was this: During the month before death the paralysed muscles were of course •resting, and thereby •losing much of their irritability and eventually becoming less irritable than the muscles on the healthy side. This gives the

A B C — a b c and

B C — b c

of the Method of Difference. When one antecedent (increased irritability) was changed and the other circumstances kept the same, the consequent didn’t follow; and when a new antecedent was provided, contrary to the first, it was followed by a contrary consequent. This has the special advantage of proving that the delay and slowing of rigor mortis don’t depend directly on the paralysis, because that was the same in both cases, but on one effect of the paralysis, namely the increased irritability—they stopped when it stopped, and were reversed when it was reversed.

Secondly: Lowering the temperature of muscles before death increases their irritability, and also delays rigor mortis and putrefaction.

It was Brown-Séquard himself who made these truths known, through experiments that conform to the Method of Difference. There’s nothing in the nature of the process that requires comment.

Thirdly: When muscular exercise is continued to exhaustion, that lessens the muscular irritability. This is a well-known truth that depends on the most general laws of muscular action and is proved by constantly repeated experiments using the Method of Difference. Now, observation has shown that if cattle are driven too hard and then killed before they recover from their fatigue, their bodies become rigid and putrefy in a surprisingly short time. The same thing has been observed in animals hunted to death, cocks killed during or shortly after a fight, and soldiers slain in battle. The only thing involving the muscles that all these have in common is their having just been subjected to exhausting exercise. Under the canon of the Method of Agreement, therefore, we can infer that there is a connection between the two facts. We have seen that the Method of Agreement can’t prove causation; but we already know that what we’re dealing with here is causation. It’s certain that the body’s state after death must somehow depend on its state at the time of death; so we are justified in concluding that the single circumstance shared by all the instances is the part of the antecedent that causes that particular consequent.

Fourthly: In proportion as the nutrition of muscles is in a good state, their irritability is high; this is supported also by laws of physiology based on many familiar applications of the Method of Difference. Now, when someone (or some animal) dies from accident or violence, with his muscles in a good state of nutrition, •the muscular irritability continues long after death, •rigor sets in late, and •it continues for a long time without putrefaction. On the other hand, in cases of disease where nutrition has been diminished for a long time before death, all these effects are reversed. This satisfies the conditions of the Joint Method of Agreement and Difference. These cases of delayed and long continued rigor agree only in being preceded by a high state of nutrition of the muscles; the cases of rapid and brief rigor agree only in being preceded by a low state of muscular nutrition; so a connection is inductively proved between •the degree of the nutrition and •the slowness and prolongation of the rigor.

Fifthly: Convulsions lessen the muscular irritability, like exhausting exercise but even more. When death follows violent and prolonged convulsions—as in tetanus, hydrophobia, some cases of cholera, and certain poisons—rigor sets in very rapidly and after a very little time gives place to putrefaction. This involves the Method of Agreement in the same way as ‘Thirdly. . . ’.

Sixthly: The last series of instances that I’ll present is more complex and requires a more finely detailed analysis.

It has long been observed that in some cases of death by lightning rigor mortis either doesn’t occur at all or doesn’t last long enough to be noticed, and that in these cases putrefaction is very rapid; whereas in other cases the usual rigor mortis appears. There must be some difference in the cause to account for this difference in the effect. [Mill reports the experimental work by Brown-Séquard that located the line between the two kinds of death by lightning (which he brought within experimental reach by substituting artificial galvanic shocks for natural lightning), namely: When and only when the eclectic shock produced muscular convulsions throughout the body, the irritability of the muscles went down, and the duration of the rigor went down with it. We can safely spare ourselves the details, and rejoin Mill when he quotes Brown-Séquard’s summing up of his findings from all the work described in this section:]

‘When the degree of muscular irritability at the time of death is considerable, either because of

•a good state of nutrition, as in persons who die in full health from an accidental cause, or

•rest, as in cases of paralysis, or

•the influence of cold,

rigor mortis sets in late and lasts long, and putrefaction appears late and progresses slowly; but when the degree of muscular irritability at the time of death is slight, either because of

•a bad state of nutrition, or

•exhaustion from overexertion, or

•convulsions caused by disease or poison,

rigor mortis sets in and ceases soon, and putrefaction appears and progresses quickly.’

These facts completely satisfy the conditions of the Joint Method of Agreement and Difference. Early and brief rigor takes place in cases that agree only in having a low state of muscular irritability. Rigor begins late and lasts long in cases that agree only in the opposite circumstance of high and unusually prolonged muscular irritability. It follows that there’s a causal connection between the degree of muscular irritability after death and the tardiness and length of the rigor mortis.

This investigation shines a strong light on the value and efficacy of the Joint Method. We have seen that the defect of that Method—as of the Method of Agreement—is that it can’t prove causation. But in the present case (as in one of the steps in the argument leading up to it) causation is already proved; because there could never be any doubt that the rigor and the ensuing putrefaction are caused by death; the empirical basis for this is too familiar to need analysis, and falls under the heading of the Method of Difference. So we know beyond doubt that the aggregate antecedent, the death, is the actual cause of the whole sequence of consequents; and ·we can get more fine-grained results—‘The death’s being of this kind is the cause of such-and-such a feature of the upshot’·—when variations in the manner of death can be shown to match corresponding variations in the effect we are investigating. . . .

§5. Some more examples: The examples I have presented offer such a clear conception of the use and practical management of three of the four methods of experimental inquiry that there’s no need to give further examples of them. There remains the Method of Residues, which hasn’t yet made an appearance in this chapter. I shall quote from Herschel some examples of that method, with the remarks by which they are introduced.

‘It is by this process that science in its present advanced state is chiefly promoted. Most natural phenomena are very complicated; and when the effects of all known causes are estimated exactly and set aside, the residual facts are constantly appearing in the form of entirely new phenomena that lead to the most important conclusions.

‘For example, the return of the comet predicted by Professor Encke a great many times in succession, and the general good agreement of its calculated place with its observed place during any one of its periods of visibility, would lead us to say that its gravitation toward the sun and planets is the sole and sufficient cause of all the facts about its orbital motion; but when the effect of this cause is strictly calculated and subtracted from the observed motion, there remains a residual phenomenon that would never have been known to exist if this method weren’t used. This residue is a small diminution of the comet’s periodic time that can’t be accounted for by gravity, and whose cause is therefore to be inquired into. Such a diminution would be caused by the resistance of a medium disseminated through the celestial regions; and as there are other good reasons for believing this to be a vera causa [see Glossary] it has therefore been ascribed to such a resistance.’ [The idea is: resistant medium → slower movement → less propulsive force relative to centripetal force → greater tendency towards the sun → shorter journey → shorter time.]

[Herschel’s next example is actually not a use of the Method of Residues, Mill says. There are several more, but we can settle for one more, introduced again by Herschel:]

‘Unexpected and striking confirmations of inductive laws frequently occur in the form of residual phenomena, during investigations that are nothing like the ones that led to the inductions themselves. An elegant example is the unexpected confirmation of the law of the development of heat in elastic fluids by compression, which is provided by the phenomena of sound. The inquiry into the cause of sound had led to conclusions about its mode of propagation, from which its velocity in the air could be precisely calculated. The calculations were performed, and the results were near enough to right to show the general correctness of the theory about the cause and the mode of propagation; but this theory couldn’t be shown to account for all the sound’s velocity. There was still a residual velocity to be accounted for, and for a long time this remained a puzzle. Eventually Laplace had the nice idea that it might come from the heat developed by the condensation that necessarily takes place at every vibration by which sound is conveyed. This matter was subjected to exact calculation, and the immediate result was the complete explanation of the residual phenomenon, and a striking confirmation of the general law of the development of heat by compression, under circumstances beyond artificial imitation.’

§6. Whewell has expressed an unfavourable opinion of the utility of the Four Methods, as well as of the aptness of the examples by which I have tried to illustrate them. He writes:

‘The obvious thing to say about these methods is that they take for granted the very thing that it’s hardest to discover, the reduction of the phenomena to formulae such as are here presented to us. When we have any set of complex facts offered to us. . . ., and we want to discover the law of nature that governs them—or, if you want to put it this way, the feature in which all the cases agree—where are we to look for our A, B, C, and a, b, c? Nature doesn’t present the cases to us in this form; and how are we to reduce them to this form? You say when we find the combination of A B C with a b c and A B D with a b d, then we may draw our inference. Granted; but when and where are we to find such combinations? Even now that the discoveries are made, who will point out to us what are the A, B, C, and a, b, c, elements of the cases that have just been enumerated? [He has cited ones from astronomy, mechanics, optics, and chemistry.] Who will tell us which of the methods of inquiry those historically real and successful inquiries exemplify? Who will carry these formulae through the history of the sciences, as they have really grown up, and show us that these four methods have been operative in their formation; or that any light is thrown upon the steps of their progress by reference to these formulae?’

He adds that in this work ·of mine· the methods haven’t been applied ‘to a large body of conspicuous and undoubted examples of discovery, extending along the whole history of science’; which ought to have been done if the methods were to be shown to have the advantage. . . .of being those ‘by which all great discoveries in science have really been made’.

These objections against the Canons of Induction are strikingly like the 18th century objections, by men as able as Whewell, against the acknowledged Canon of Ratiocination. Those who protested against the Aristotelian logic said of the syllogism what Whewell says of the inductive methods, namely that it ‘takes for granted the very thing that is most difficult to discover, the reduction of the argument to formulae such as are here presented to us’. The great difficulty, they said, is to obtain your syllogism, not to judge its correctness when obtained. On the matter of fact, they and Whewell are right. The greatest difficulty in both cases is •obtaining the evidence and then •reducing it to the form that tests its conclusiveness. But if we try to reduce it without knowing what it’s to be reduced to we’re not likely to make much progress. It’s harder to solve a geometrical problem than to judge whether a proposed solution is correct; but if people couldn’t judge the solution when it was found, they would have little chance of finding it. And it can’t be maintained that to judge an induction once it has been found is perfectly easy, a thing for which aids and instruments are superfluous; for erroneous inductions, •false inferences from experience, are quite as common as—and on some subjects much commoner than—•true ones. The business of inductive logic is to provide rules and models (such as the syllogism and its rules are for ratiocination) such that inductive arguments are conclusive if, and only if, they conform to them. That’s what the four methods claim to be, and what I believe they are considered to be by all experimental philosophers, who had practised all of them long before anyone tried to reduce the practice to theory.

The assailants of the syllogism also anticipated Whewell in the other branch of his argument. They said that no discoveries were ever made by syllogism; and Whewell seems to say, that none were ever made by the four methods of induction. To the former objectors Whately gave a good answer, namely that if their argument was any good it was good against the reasoning process altogether; for whatever can’t be reduced to syllogism isn’t reasoning. And Whewell’s argument, if good at all, is good against all inferences from experience. In saying that no discoveries were ever made by the four methods, he affirms that none were ever made by observation and experiment; for assuredly if any were, it was by processes reducible to one or other of those methods.

This difference between us accounts for his dissatisfaction with my examples, which I didn’t select with a view to showing that observation and experiment are ways of acquiring knowledge. In choosing them I was thinking only of •illustration, and of •making methods easier to grasp by examples. If I had wanted to justify the processes themselves as means of investigation, I wouldn’t have needed to look far off or use recondite or complicated instances. As a specimen of a truth ascertained by the Method of Agreement, I could have chosen the proposition ‘Dogs bark’. This dog and that dog and the other dog answer to A B C and A D E and A F G. The circumstance of being a dog answers to A. Barking answers to a. As a truth made known by the Method of Difference, ‘Fire burns’ might have sufficed. Before I touch the fire I am not burned; this is B C; I touch it, and am burned; this is A B C and a B C.

Whewell doesn’t regard such familiar experimental processes as inductions; but they are perfectly homogeneous with the ones on which, even on his own showing, the pyramid of science is based. He tries to escape from this conclusion by arbitrarily restricting the range of examples that can serve as instances of induction: they must not be

•things that are still matters of discussion,

•drawn from mental and social subjects, or

•drawn from ordinary observation and practical life.

They must all concern generalisations by which scientific thinkers have ascended to great and comprehensive laws of natural phenomena. Now it is seldom possible in these complicated inquiries to go much beyond the first steps without making use of deduction and the temporary aid of hypotheses—this being something that Whewell and I have maintained against the purely empirical school—so that such cases wouldn’t serve well as illustrations of the principles of mere observation and experiment. Whewell is misled by their absence into representing the experimental methods as serving no purpose in scientific investigation, forgetting that if those methods hadn’t supplied the first generalisations there would have been no materials for his own conception of induction to work on.

But it’s easy to answer his challenge to say which of the four methods are involved in certain important scientific developments. The planetary paths, as far as they are a case of induction at all [see here], involves the Method of Agreement. The law of ‘falling bodies’, namely that they cover distances proportional to the squares of the times, was historically a deduction from the first law of motion; but the experiments that verified it and could have led to its discovery involved the Method of Agreement; and the apparent variation from the true law caused by air-resistance was cleared up by experiments in vacuo, involving the Method of Difference. . . . The movements of comets were determined by highly complex processes of thought in which deduction was predominant, but the Methods of Agreement and of Concomitant Variations had a large part in establishing the empirical laws. Every case. . . .is a well-marked example of the Method of Difference. To anyone acquainted with the subjects—to Whewell himself—there wouldn’t be the slightest difficulty in setting out ‘the A B C and a b c elements’ of these cases.

If discoveries are ever made by observation and experiment without deduction, the four methods are methods of •discovery; but even if they weren’t, they would still be the sole methods of •proof; and they could serve as proofs even of the results of deduction. The great generalisations that begin as hypotheses must end by being proved, and in due course I’ll show that they are in fact proved by the four methods. Now logic is principally concerned with proof as such. This approach has no chance of finding favour with Whewell, because his system has the special feature that it doesn’t recognise any need for proof in cases of induction. If an hypothesis is carefully collated with facts, and nothing inconsistent with it turns up—i.e. if experience doesn’t disprove it—Whewell is content, at least until we find a simpler hypothesis that is equally consistent with experience. If this is induction, doubtless there is no need for the four methods. But to suppose that it is induction seems to me a radical misunderstanding of the nature of the evidence for physical truths.

There’s a real practical need for a test for induction, like the syllogistic test of ratiocination. Inferences that defy the most elementary notions of inductive logic are confidently presented by persons eminent in physical science, as soon as they are off the factual ground that they know. . . . As for educated persons in general, I doubt that they are better judges of a good or a bad induction than they were before Bacon wrote. The improvement in the •results of thinking has seldom extended to the •processes; and if it has reached any process it has been that of investigation only and not that of proof. No doubt a knowledge of many laws of nature has been arrived at by forming hypotheses and finding that the facts corresponded to them; and many errors have been cured by coming to know facts that were inconsistent with them, but not by discovering that the mode of thought that led to the errors was itself faulty and could have been known to be faulty independently of the facts that disproved the specific conclusion. The upshot is that while mankind’s thoughts on many subjects have worked out well in practice, the thinking power remains as weak as ever. In all subjects where the facts that would check the result are not accessible—e.g. in what relates to the invisible world, and even. . . .to the visible world of the planetary regions—men with the greatest scientific acquirements argue as pitiably as the merest ignoramus. They have made many sound inductions, but they haven’t learned from them—and Whewell thinks there is no need for them to learn—the principles of inductive evidence.

NOTES

[1] For this bit of theorising as for many of my other scientific illustrations I am indebted to Bain, whose treatise on Logic is full of apt illustrations of all the inductive methods.

[2] This example may seem to count against my claim that the Method of Difference doesn’t apply well to cases of pure observation ·as distinct from controlled experiments·; but really it doesn’t. Nature seems to have imitated man’s type of experiment, but has succeeded only in copying man’s most imperfect experiments—namely, those in which he succeeds in producing the phenomenon only by using complex •means that he can’t perfectly analyse and therefore can’t tell what parts of the effects may be due not to the supposed cause but to some unknown agency of the •means by which that cause was produced. In the natural experiment in question here, the •means was the clearing off a canopy of clouds; and we don’t know enough about this process. . . .to be certain a priori that it couldn’t operate upon the deposition of dew independently of any effect on the temperature of the earth’s surface. Thus, even in a case as favorable as this to Nature’s experimental talents, her experiment is of little value except in corroboration of a conclusion already reached through other means.

Chapter 10. Plurality of causes, and the intermixture of effects

§1. In my account of the four methods of observation and experiment by which we contrive to sort out among a mass of coexistent phenomena the particular effect of a given cause, or the particular cause of a given effect, I have had to suppose for simplicity’s sake that this analytical operation doesn’t run into difficulties other than the ones that are essentially inherent in its nature. So I have represented every effect as connected exclusively with a single cause, and as incapable of being confusingly mixed in with any other coexistent effect. I have regarded a b c d e, the aggregate of the phenomena existing at any moment, as consisting of dissimilar facts—a and b and c and d and e—for each of which we need to look for just one cause; the difficulty being only that of singling out this one cause from the multitude of antecedent circumstances A, B, C, D, and E. The cause may indeed not be simple; it may consist of an assemblage of conditions; but I have supposed that there’s only one possible assemblage of conditions from which the given effect could result.

If that were right, it would be comparatively easy to investigate the laws of nature. But the supposition is false in both its parts. (i) It’s not true that the same phenomenon is always produced by the same cause; the effect a may sometimes arise from A, sometimes from B. (ii) And the effects of different causes are often not dissimilar but homogeneous, and not demarcated by any assignable boundaries; A and B, instead of producing a and b, may produce different parts of an effect a. Investigation of the laws of phenomena is made much harder and darker by the need to take account of these two circumstances: intermixture of effects, and plurality of causes. I’ll take the latter first, because it is the simpler of the two. ·I’ll start on the intermixture of effects in section 4·.

·Here’s the situation that we face·. It’s not true that one effect must be connected with only one cause or assemblage of conditions; i.e. that each phenomenon can be produced in only one way. There are often several independent ways in which the same phenomenon could have originated. . . . Many causes can produce mechanical motion; many causes can produce some kinds of sensation; many causes can produce death. It can happen that a given effect was produced by a certain cause but could perfectly well have been produced without it.

§2. One of the principal consequences of this fact of plurality of causes is to bring uncertainty into the Method of Agreement. I illustrated that method by supposing two instances:

•A B C followed by a b c, and

•A D E followed by a d e.

To avoid a difficulty that isn’t relevant to my present theme, let us suppose that we know for sure that the two cases have no antecedent in common except A. Then it might seem that we have a basis for concluding that A is an invariable antecedent of a, and even that it is its unconditional invariable antecedent, i.e. its cause. But the moment we admit the possibility of a plurality of causes, that conclusion fails. Why? Because it tacitly assumes that a must have been produced in both instances by the same cause. If there could have been two causes, they might have been (for example) C and E; with C causing a in the former of the instances and E in the other, and A having no influence in either case.

Suppose we investigate the circumstances of the upbringing and history of two great artists (or it could be two great philosophers, two extremely selfish men, or two extremely generous men) and find that their antecedents agree only in one circumstance x; would it follow that x was what caused each to be a great artist (or a great philosopher or. . . )? Not at all! The causes that can produce any type of character are very numerous; and the two persons could have been just as alike in character without there being any resemblance between their previous histories.

This is a characteristic imperfection of the Method of Agreement, from which the Method of Difference is free. For if we have two instances A B C and B C, of which B C gives b c, and the addition of A converts it into a b c, it’s certain that at least in this instance A was either the cause of a or an indispensable portion of its cause, even if in other instances a is produced by entirely different causes. Plurality of causes, therefore, doesn’t make the Method of Difference less reliable, and doesn’t even require a greater number of observations or experiments; two instances, one positive and the other negative, are still enough for a complete and rigorous induction. Not so with the Method of Agreement. The conclusions that it yields when the number of instances is small are of no real value unless they function as suggestions that may lead either to •experiments bringing them to the test of the Method of Difference or •to reasonings that can explain and verify them deductively.

When the instances are indefinitely multiplied and varied and still suggest the same result, then (and only then) we have an independently valuable result. If the only instances ·of production of a· are A B C and A D E, though these instances have nothing in common except A, the effect a may have been produced in the two cases by different causes so that there’s at most only a slight probability in favour of A; there may be causation but it’s almost equally probable that there was only a coincidence. But the oftener we repeat the observation, varying the circumstances, the more we advance toward a solution of this doubt. For if we try A F G, A H K, etc., all unalike except in containing A, and if we find the effect a appears in all these cases, we must suppose one of two things: •that a is caused by A, or •that it has as many different causes as there are instances. With each addition, therefore, to the number of instances, the presumption is strengthened in favour of A. The inquirer will take any chance he gets to exclude A from one of these combinations—let’s say from A H K—and by trying H K separately bring the Method of Difference to the aid of the Method of Agreement. Only the Method of Difference can show us that A is the cause of a; but the Method of Agreement, provided the instances are numerous and sufficiently various, can put it beyond any reasonable doubt that A is either the cause of a or an effect of the cause of a.

How many varied instances with only A in common does it take to •rule out the supposition of a plurality of causes and •make it virtually certain that a is connected with A? We mustn’t dodge this question, but the consideration of it belongs to the theory of probability, which I’ll come to in chapter 17. Still, we can see right away •that the conclusion does amount to a practical certainty after a sufficient number of instances, and thus •that the method isn’t radically discredited by the characteristic imperfection. There are two upshots to these considerations, (1) We see a new source of inferiority in the Method of Agreement, and new reasons for never resting content with results obtained by it without trying to confirm them either by the Method of Difference or by connecting them deductively with some law already ascertained by that superior method. (2) We learn the true theory of the value of mere number of instances in inductive inquiry. The plurality of causes is the only reason why mere number is of any importance. Unscientific inquirers tend to rely too much on number, without analysing the instances—without looking into their nature closely enough to see what circumstances are or aren’t eliminated by means of them. Most people hold their conclusions with a degree of assurance proportioned to the mere mass of the experience they appear to rest on, overlooking the fact that by adding instances to instances, differing from one another only in details already recognised as immaterial, nothing whatever is added to the force of the conclusion. A single instance eliminating some antecedent that existed in all the other cases is of more value than the greatest multitude of instances that are reckoned by their number alone. We do of course have to assure ourselves, by repetition of the observation or experiment, that we haven’t committed any error concerning the individual facts observed; and until we are sure about this our primary need is not to vary the circumstances but to repeat the same experiment or observation, very carefully, without any change. But once we have this assurance, the multiplication of instances that don’t exclude any more circumstances is entirely useless, provided there have been already enough to exclude the supposition of plurality of causes.

This is important: . . . .the Joint Method of Agreement and Difference is not affected by the characteristic imperfection of the Method of Agreement. In the joint method it is supposed not only that •the instances in which a is ·an effect· agree only in containing A, but also that •the instances in which a is not ·an effect· agree only in not containing A. If that’s how things stand, A must be not only •the cause of a but •the only possible cause; for if there were another—say, B—then in the instances in which a is not ·an effect· B must have been absent as well as A, and it wouldn’t be true that these instances agree only in not containing A. This is an immense advantage of the •joint method over the •simple Method of Agreement. It may seem, indeed, that the advantage belongs to the negative part of the joint method rather than to the method as a whole. The Method of Agreement, when applied to negative instances (i.e. ones where a phenomenon does not take place), is certainly free from the characteristic imperfection which affects it in the affirmative case. So you might think that the negative premise could be worked as a simple case of the Method of Agreement, with no need for an affirmative premise to go with it. But though this is true in principle, it’s usually impossible to work the Method of Agreement by negative instances without positive ones, because it’s so much harder to exhaust the field of negation than the field of affirmation. For example: if we are inquiring into what makes bodies transparent, what are our chances of success if we try to discover what the various substances that aren’t transparent have in common. We are more likely to succeed in seizing some point of resemblance among the comparatively few and definite kinds of things that are transparent; and when we’ve done this our natural next task is to look into whether the absence of this one circumstance isn’t precisely the respect in which all opaque substances will be found to be alike.

So the Joint Method of Agreement and Difference. . . .is, after the Direct Method of Difference, the most powerful of the instruments of inductive investigation that I haven’t yet discussed; and in the sciences that depend on pure observation with little or no aid from experiment, this method—so well illustrated by the theorising about the cause of dew—is the primary resource, so far as direct appeals to experience are concerned.

§3. Up to here I have treated plurality of causes only as a possible supposition that makes our inductions uncertain until we have eliminated it; and have considered how we can eliminate it in cases where there isn’t in fact a plurality of causes. But we must also consider it as something that actually occurs in nature, and find ways for our methods of induction to be able to identify the cases where it does occur. We don’t need any special method for doing this. When an effect really could be produced by either of two (or more) causes, the process for detecting them is exactly the same as the process for discovering single causes. They may (first) be discovered as separate sequences, by separate sets of instances—i.e. of observations or experiments—showing that the causes of heat include

•the sun,

•friction,

•percussion,

•electricity,

•chemical action,

with each of these being shown by its own special set of instances. Or (secondly) the plurality may come to light when we are collating a number of instances in an attempt to find something that they all have in common. A failing attempt: we can’t find anything that is common to all instances of heat; we find that no one antecedent is present in all the instances, no one of them indispensable to the effect. But when we look harder we find that although no one is always present, one or other of several always is. If on further analysis we can detect any common element in these, we may be able to ascend from them to some one cause that is the really operative circumstance in them all. Thus it is now thought that a single ultimate source is at work in the production of heat by friction, percussion, chemical action, etc. But if (as continually happens) we can’t take this further step, the different antecedents must be noted provisionally as distinct causes each of which is sufficient, unaided, to produce the effect.

I now move from the plurality of causes to the still more special and more complex case of •the intermixture of effects and •the interference of causes with one another. This is the principal source of complication and difficulty in the study of nature; and we’ll soon see that the four inductive methods that I have presented—the only possible methods of directly inductive investigation by observation and experiment—are for the most part quite unable to cope with it. Our only means for unravelling the complexities proceeding from the intermixture of effects and the interaction amongst causes is deduction; and the four methods can’t do much more than supply premises for our deductions and check their conclusions.

§4. A concurrence of two or more causes, not separately producing each its own effect but interfering with or altering one another’s effects, happens in two ways. (1) In one, exemplified by the joint operation of different forces in mechanics, the separate effects of all the causes continue to be produced, but are compounded with one another and make one total. (2) In the other, already mentioned here and illustrated by the case of chemical action, the separate effects cease entirely and are succeeded by phenomena that are altogether different and governed by different laws.

Of these (1) is by far the more frequent, and also the more likely to elude the grasp of •our experimental methods. The exceptional case (2) is basically open to being handled by •them. When

•the laws of the original agents cease to be applicable because a new phenomenon appears that doesn’t offer a hand-hold for those laws, e.g. when

•two gaseous substances, hydrogen and oxygen, are brought together and throw off their special properties and produce water,

in such cases the new fact can be subjected to experimental inquiry, like any other phenomenon; and the elements that are said to compose it can be regarded as the mere agents of its production—the conditions on which it depends, the facts that make up its cause.

The effects of the new phenomenon—e.g. the properties of water—are as easily found by experiment as the effects of any other cause. But to discover the cause of it, i.e. the particular conjunction of agents from which it results, is often difficult enough. (a) The origin and actual production of the phenomenon are usually out of reach of our observation. If we couldn’t have learned the composition of water until we found instances where it was actually produced from oxygen and hydrogen, we’d have been forced to wait until someone had the random idea of passing an electric spark through a mixture of the two gases, or inserting a lighted taper into it, merely to see what would happen. (b) Many substances can be analysed but can’t be recompounded by any known artificial means. (c) Even if we could have learned by the Method of Agreement that oxygen and hydrogen are both present when water is produced, no experiments with oxygen and hydrogen separately—no knowledge of their ·separate· laws—could have enabled us to infer deductively that they would produce water. For that we need a specific experiment on the two combined.

Given these difficulties, you might expect that our knowledge of the causes of this class of effects comes either from accident or from the gradual progress of experimentation on the different combinations that the producing agents are capable of. But ·we can often do better than that, because· effects of this kind have the special feature that under certain combinations of circumstances they reproduce their causes. Water results from putting hydrogen and oxygen really closely and intimately together, and correspondingly hydrogen and oxygen result from placing water in certain situations. When that happens the new laws—·i.e. the laws of water·—abruptly cease and the two gases re-appear separately with their own properties. What is called ‘chemical analysis’ is the process of searching for the causes of a phenomenon among its effects, or rather among the effects of subjecting it to some other causes.

Lavoisier, by heating mercury to a high temperature in a closed vessel containing air, found that the mercury increased in weight and became ‘red precipitate’, while the air when examined after the experiment turned out to have •lost weight and •lost its ability to support life or combustion. When red precipitate was exposed to a still greater heat it became mercury again and gave off a gas which did support life and flame. . . .

Where two phenomena between whose laws or properties no connection can be traced are thus cause-effect and effect-cause, each capable in its turn of being produced from the other, and each when it produces the other ceases itself to exist (as water is produced from oxygen and hydrogen, and oxygen and hydrogen are reproduced from water); this causation of the two phenomena by one another, each being generated by the other’s destruction, is strictly transformation. The idea of chemical composition is an idea of transformation, but of a transformation that is incomplete because we consider the oxygen and hydrogen to be present in the water as oxygen and hydrogen, discoverable in it if our senses were keen enough. That’s a mere a supposition, based solely on the fact that the weight of the water is the sum of the separate weights of the two ingredients. This fact about the weights is an exception to the entire disappearance in the compound of the laws of the separate ingredients. . . . In these cases, where the heteropathic effect (as I called it here) is merely a transformation of its cause—i.e. where an effect and its cause are also a cause and its effect, and are mutually convertible into each other—the problem of finding the cause resolves itself into the far easier one of finding an effect, which is the kind of inquiry that can be performed by direct experiment. But in some cases of heteropathic effects this can’t be done. Consider for instance the heteropathic laws of mind; the part of the phenomena of our mental nature that are analogous to •chemical rather than to •dynamical phenomena; as when a complex passion is formed by the coalition of several elementary impulses, or a complex emotion by several simple pleasures or pains of which it is the result without being the aggregate or in any way homogeneous with them. in these cases the product is generated by its various factors, but the factors can’t be reproduced from the product. . . . We can’t ascertain what simple feelings any of our complex states of mind are generated from, in the way we ascertain the ingredients of a chemical compound by making it generate them. So our only way to discover these laws is the slow process of studying the simple feelings themselves, and learning by experimenting on the various combinations they’re capable of, what they can generate by their interactions.

§5. One might have thought that the other and apparently simpler sort of the mutual interference of causes, where each cause continues to obey the laws that it conformed to in its separate state, would have presented fewer difficulties to the inductive inquirer than does the one I have been discussing. In fact, however, it presents—so far as direct induction without help from deduction is concerned—infinitely greater difficulties. When a concurrence of causes gives rise to a new effect that has no relation to the separate effects of those causes, the resulting phenomenon stands out undisguised. . . ., presenting no obstacle to our recognising its presence or absence among any number of surrounding phenomena. So it can easily be brought under the canons of induction if instances of the right kind can be obtained. And the non-occurrence of such instances, or the lack of means to produce them artificially, is the real and only difficulty in such investigations—not a logical difficulty but in some way a physical one. What I (in chapter 5) called the ‘composition of causes’ is not like that. There, the effects of the separate causes don’t terminate and give place to others, thereby ceasing to form any part of the phenomenon to be investigated; on the contrary, they hold their place but are intermingled with, and disguised by, the homogeneous and closely allied effects of other causes. They are no longer

a, b, c, d, e

existing side by side and continuing to be separately discernible; rather, they are

+a, -a, ½b, -b, 2b, etc..

some of which cancel one another, while many others don’t appear separately but merge in one sum. Between •their over-all result and •the causes by which it was produced there’s often an insurmountable difficulty in seeing any fixed relation whatever.

We have seen that according to the general idea of the composition of causes:

Two or more laws interfere with one another, apparently frustrating or modifying one another’s operation, yet in reality all are fulfilled, the collective effect being the exact sum of the effects of the causes taken separately.

As a familiar example, think about a body in equilibrium by two equal and opposite forces. One on its own would carry the body in an hour one mile westward, the other on its own would carry it in an hour one mile eastward; and the result ·of the equilibrium· is just the same as if the body had first been carried westward by one force and then back eastward by the other, being finally left where it was at first.

Every causal law L_1 is liable to be counteracted— seemingly frustrated—by coming into conflict with another law L_2 (or more than one) the separate result of which is opposite to L_1’s, more or less inconsistent with it. The result of that is that many instances in which L_1 really is entirely fulfilled don’t at first sight seem to involve its operation at all. An example of that is the west-east one that I just offered: a ‘force’ in mechanics means precisely a ‘cause of motion’, but it can happen that the sum of the effects of two causes of motion is rest ·= motionlessness·. Another example: a body subjected to two forces in different directions moves in the diagonal; and it seems a paradox to say that motion in the diagonal is the sum of two motions in two other lines. [The ‘diagonal’ referred to here is, as Mill explained here, the diagonal of a parallelogram whose sides represent the direction and strength of those two forces.] Motion, however, is merely change of place, and at every instant the body is in the exact place it would have been in if the forces had acted during alternate instants instead of acting in the same instant (except that we must of course allow the forces double the time if they’re to do successively what they in fact do simultaneously). So it’s clear that each force has had during each instant its own full effect, and that the modifying influence that C_2 is said to exercise with respect to C_1 can be seen as exerted not over the action of C_1 itself but over the effect after C_1 has done its work. For all purposes of predicting, calculating, or explaining their joint result, causes that compound their effects can be treated as if •they produced their own separate effects simultaneously, and •all these effects coexisted visibly.

Because the laws of causes are just as completely fulfilled •when the causes are ‘counteracted’ by opposing causes as they are •when they are left to their own undisturbed action, we must take care not to express the laws in terms that would make the assertion of their being fulfilled in those cases a contradiction. For example, if we said that there’s a law of nature according to which

any body to which a force is applied moves in the direction of the force with a velocity directly proportional to the force and inversely proportional to its own mass,

when in fact

some bodies to which a force is applied don’t move at all, and the ones that do move (at least in the region of our earth) are from the very first held back by the action of gravity and other resisting forces, and eventually stopped altogether,

it’s clear that the general proposition—·the supposed ‘law’·— though true under a certain hypothesis doesn’t express the facts as they actually occur. To get the expression of the law to fit the real phenomena we must say not that the object moves in the direction and with the velocity specified but that it tends to move in that way. We could guard our expression in a different way by saying that the body moves in that manner unless prevented, or except to the extent that it is prevented, by some counteracting cause. But ·that’s less good, because· the body doesn’t merely

move in that manner unless counteracted;

it also

tends to move in that manner even when it is counteracted.

It still exerts, in the original direction, the same energy of movement as if its first impulse had been undisturbed, and produces by that energy an exactly equivalent quantity of effect. This is true even when the force leaves the body as it found it, in a state of absolute rest. Suppose we are trying to raise a body weighing three tons with a force equal to one ton; if while we’re doing this wind or water or any other agent supplies an additional force of just over two tons, the body will be raised—proving that the force we applied exerted its full effect by neutralizing an equivalent part of the total weight. And if, while we’re exerting this force of one ton on the object in a direction contrary to that of gravity, it is put onto a scale and weighed, it will be found to have lost a ton of its weight—i.e. to press downward with a force equal to only the difference between the two forces.

These facts are correctly indicated by the term ‘tendency’. Because laws of causation can be counteracted, they should all be stated in terms of tendencies only, not actual results. . . .

The habit of neglecting this needed element in the precise expression of the laws of nature has given rise to the popular [see Glossary] prejudice that all general truths have exceptions; and this has brought much unmerited distrust to the conclusions of science when they have been submitted to the judgment of minds insufficiently disciplined and cultivated [see Glossary]. The rough generalisations suggested by common observation usually do have exceptions; but principles of science—i.e. laws of causation—don’t. Let me quote from an earlier work of mine [from here to the end of this section]: ‘What is thought to be an exception to a principle is always some other principle cutting into the former, some other force that impinges against the first force and pushes it off-course. We do not have this:

a law and an exception to it, the law acting in 99 cases and the exception in one.

What we do have is this:

Two laws, each possibly acting in the whole 100 cases, bringing about a common effect by their joint operation.

If in a single case the less conspicuous force—called the “disturbing” force—prevails sufficiently over the other force to create what is commonly called an “exception”, the same disturbing force probably acts as a modifying cause in many other cases that no-one will call exceptions.

‘Thus if it were said to be a law of nature that all heavy bodies fall to the ground, it would probably be said that the resistance of the atmosphere that prevents a balloon from falling makes the balloon an “exception” to that supposed law of nature. But the real law is that all heavy bodies tend to fall; and there are no exceptions to this, not even the sun and moon, because (as every astronomer knows) they tend toward the earth with a force exactly equal to that with which the earth tends toward them. . . .’ [From Mill’s Essays on Some Unsettled Questions of Political Economy, Essay 5.]

§6. We now have to face the question: How are we to study these complex effects made up of the effects of many causes? What enable us to trace each effect back to the concurrence of causes in which it originated, and learn what the circumstances are in which it may be expected to recur?

The conditions of a phenomenon that arises from a composition of causes can be investigated either •deductively or •experimentally.

It’s obvious that the deductive mode of investigation is appropriate to this kind of case. The law governing an effect of this sort, x, is an upshot of the separate laws governing the causes that jointly produced x, so it is in itself capable of being deduced from these laws. This is called the a priori method. The a posteriori method claims to proceed according to the canons of experimental inquiry. Considering the whole assemblage of causes that jointly produced x as one single cause, it tries to ascertain the cause in the ordinary manner, by a comparison of instances. There are two varieties of this second method. If it merely assembles and compares instances of the effect, it’s a method of pure observation. If it operates on the causes and tries different combinations of them in hopes of eventually hitting the precise combination that will produce the given total effect, it is a method of experiment.

So we have three methods: deductive, observational, and experimental. In order get clearer about the nature of each, and determine which of them deserves preference, I shall ‘clothe them in circumstances’ (Lord Eldon’s phrase). I’ll select for this purpose a case that hasn’t yet provided a brilliant example of the success of any of the three methods, but does illustrate the difficulties inherent in them. Let’s suppose we are inquiring into. . . .the conditions of recovery from a given disease x; and for a start let our question be limited to: ‘Is mercury a remedy for x?’

•The deductive method would set out from known properties of mercury and known laws of the human body, and by reasoning from these would try to discover whether mercury will act on an x-afflicted body in a manner that would tend to restore health. •The experimental method would simply administer mercury in as many cases as possible, noting the age, sex, temperament, and other details of bodily constitution, the particular form or variety of x, the particular stage of its progress etc., noting in which of these cases it led to a salutary effect, and what circumstances it was combined with on those occasions. •The method of simple observation would compare instances of recovery, to find whether they agreed in having been preceded by the administration of mercury; or would compare instances of recovery with instances of failure, to find cases that were like the others except that in them mercury had been administered, or. . . .that it hadn’t.

§7. No-one has ever seriously contended that the last of these three methods of investigation could work with composite causes. No useful conclusions on a subject of such intricacy were ever obtained in that way. The most one could get would be •a vague general impression for or against the efficacy of mercury, and •that would be practically useless unless it were confirmed by one of the other two methods. The results that this method tries to obtain would be of the utmost possible value if they could be obtained. If in an examination of a great number of instances all recoveries were cases in which mercury had been administered, we could generalise with confidence from this experience, and would have obtained a conclusion of real value. But in a case of this sort we have no chance of getting a basis for such a generalisation. Why not? because of what here I called the ‘characteristic imperfection’ of the Method of Agreement, namely the plurality of causes. Even if mercury does tend to cure the disease, so many other natural and artificial causes also tend to cure it that there are sure to be abundant instances of recovery in which mercury has not been administered. . . .

When an effect results from the union of many causes, no one of them can have a large role in determining •whether the effect follows or, if it does, •what it is like in detail. Recovery from a disease is an event that always comes from many influences acting together. Mercury may be one such influence; but there are bound to be cases where it is administered but the patient doesn’t recover because other needed influences aren’t at work, or where mercury isn’t given but the patient recovers because the other favourable influences are powerful enough to do this in its absence. . . . About the best that this method could do for us is to show—by multiplied and accurate hospital records and the like—that there are rather more recoveries and rather fewer failures when mercury is administered than when it isn’t. But that result would have little value as a guide to practice, and virtually none as a contribution to the theory of the subject.[1]

§8. Having recognised the inapplicability of the method of simple observation to ascertain the conditions of effects that have many concurring causes, let us now ask whether any greater benefit can be expected from the other branch of the a posteriori method—the one that directly tries different combinations of causes. . . .and takes note of their effects; e.g. trying the effect of mercury in as many different circumstances as possible. This method differs from the previous one in turning our attention directly to the causes or agents, instead of turning it to the effect, recovery from the disease. As a general rule the effects of causes are easier to study than the causes of effects, so it’s natural to think that this method has a better chance of succeeding than the previous one.

The method now under consideration is called the Empirical Method; and to estimate it fairly we must take it to be completely empirical, without any input from any deductive operation. We might do this:

Try experiments with mercury on a healthy person in order to ascertain the general laws of its action upon the human body, and then reason from these laws to determine how it will act on persons affected with a particular disease,

and this might be a really effectual method; but it is deduction. The experimental method doesn’t derive the law of a complex case from the simpler laws that jointly produce it, but experiments directly on the complex case. We must set aside entirely all knowledge of the simpler tendencies of mercury in detail. Our experimentation must try to get a direct answer to the specific question: ‘Does mercury tend to cure the particular disease or doesn’t it?’

Let us see how far the rules of experimentation that have to be followed in other cases can be followed in the ·multiple-cause· case. [Mill’s handling of this question is long, detailed and demanding, and leads him to conclude that the rules can’t be obeyed in this ‘case’. At every turn we encounter possibilities of error that we can’t exclude because of the complexities of the multiple-cause situation. This sums it up:]

Anything like a scientific use of the method of experiment in these complicated cases is out of the question. We can generally, even in the most favourable cases, only discover by a series of trials that a certain cause is very often followed by a certain effect. Anything like a scientific use of the method of experiment is therefore out of the question in these complicated cases. Even in the most favorable cases we can generally only discover by a series of trials that a certain cause is very often followed by a certain effect. . . .

If so little can be done by the experimental method to understand multiple-cause situations in medical science, still less is this method applicable to a class of phenomena even more complicated than those of physiology—the phenomena of politics and history. In that region plurality of causes exists in almost boundless excess, and most effects are inextricably interwoven with one another. To make things still worse, most inquiries in political science relate to the production of very large-scale effects such as •the public wealth, •public security, •public morality and the like; and these items are open to being affected—directly or indirectly, helped or hindered—by nearly every fact that exists or event that occurs in human society. The vulgar notion that the safe methods on political subjects are those of Baconian induction—that the true guide is not general reasoning but specific experience—will some day be referred to as one of the clearest signs of a low state of theoretical thinking in any age in which it is accepted! Nothing can be more ludicrous than the parodies of experimental reasoning that we encounter not only in popular discussion but also in solemn treatises about the affairs of nations:

•‘How can an institution be bad, when the country has prospered under it?’

•‘How can such-and-such a cause have contributed to the prosperity of one country, when another has prospered without it?’

Anyone who argues like this, not intending to deceive, should be sent back to learn the elements of some one of the easier physical sciences! Such reasoners ignore the fact of plurality of causes in the very case that provides the most obvious example of it. [Mill adds that there’s little ‘reason for regret’ in our inability to perform experiments in this area; because even if we could perform them, we would be comprehensively defeated by the scope and complexity of the material.] The nearest approach to an experiment—in the philosophical [here = ‘scientific’] sense of the term—in politics is the introduction of a new operative element into national affairs by some particular identifiable measure of government, such as the enactment or repeal of a particular law. But where there are so many influences at work, it takes time for the influence of a new cause on national phenomena to become apparent; and as the causes operating in such an extensive sphere are not only •infinitely numerous but •in a state of perpetual alteration, it is always certain that before the effect of the new cause becomes conspicuous enough to be a subject of induction, many of the other influencing circumstances will have changed, wrecking the experiment.[2]

Thus, two of the three possible methods for the study of phenomena resulting from a combination of many causes are, from the very nature of the case, inefficient and illusory. That leaves us with the third—the method that considers the causes separately and infers the effect from the balance of the different tendencies that produce it; i.e. the deductive or a priori method. A detailed consideration of this intellectual process requires a chapter to itself.

NOTES

[1] Bain rightly says that though the Methods of Agreement and Difference are not applicable to these cases, the Method of Concomitant Variations is of some use with them: ‘If a cause happens to vary alone, the effect will also vary alone: a cause and effect may be thus singled out under the greatest complications. . . ’ [Mill says that this is correct in theory, but:] when there are many influencing causes, no one of them greatly predominating over the rest, and especially when some of them are continually changing, it is scarcely ever possible to trace a relation between •the variations of the effect and •those of any one cause in a way that would enable us to assign to that cause its real share in the production of the effect.

[2] Though Bain generally agrees with the views expressed in this chapter, he seems to estimate more highly than I do the scope for specific experimental evidence in politics. He is right when he says that there are some cases ‘when an agent suddenly introduced is almost instantaneously followed by some other changes, as when the announcement of a diplomatic rupture between two nations is followed the same day by a derangement of the money-market’. But this ‘experiment’ would be quite inconclusive merely as an experiment. It can only serve, as any experiment can, to verify the conclusion of a deduction. Unless we already knew from our knowledge of business men’s motives that the prospect of war tends to derange the money-market, we would never have been able to prove a connection between those two facts. What if we ascertained throgh historical study that one followed the other in a great number of instances? Anyone who has carefully examined any of the attempts—they’re continually being made—to prove economic doctrines by such a recital of instances knows very well how futile they are. It turns out that the circumstances of the cases have hardly ever been fully stated, and that the records have omitted as many or even more instances that would have tended to an opposite conclusion.

Chapter 11. The deductive method

§1. Given that the direct methods of observation and experiment can’t help us to grasp the conditions and laws of recurrence of the more complex phenomena, our main source of knowledge of those phenomena has to be the Deductive Method. It consists of three operations; •direct induction, •ratiocination, and •verification.

First operation, Induction: I call the first step in the process an ‘inductive’ operation, because there must be a direct induction as the basis of the whole. The role of the induction may often be played by •a prior deduction, but the premises of •this must have been derived from induction.

The problem of the Deductive Method is to find the law of an •effect from the laws of the different tendencies of which •it is the joint result. What is needed first, then, is to know the laws of those ·separate· tendencies, i.e. the law of each of the concurrent causes; and this requires, for each cause separately, a previous process of observation or experiment or else a previous deduction whose ultimate premises come from observation or experiment. If we’re investigating social or historical phenomena, the premises of the Deductive Method must be the laws of the causes that determine such phenomena; and those causes are •human actions together with •the general external circumstances by which mankind are influenced. . . . So the Deductive Method as applied to social phenomena must start by investigating. . . .the laws of human action, and the properties of external things that determine the actions of human beings in society. Some of these general truths will be obtained by observation and experiment, others by deduction (e.g. deducing the more complex laws of human action from the simpler ones); but the simple or elementary laws must have been obtained by a directly inductive process.

Learning what the separate causes are that must be studied in this way is sometimes hard, sometimes easy. In the social-phenomena case it is easy. There could never have been any doubt that social phenomena depend on the acts and mental impressions of human beings, however little may have been known about what laws govern those impressions and actions, or what social consequences their laws naturally lead to. ·Another easy case:· After physical science had achieved a certain development, there was no real doubt about where to look for the laws on which the phenomena of life depend: they had to be the mechanical and chemical laws of the solid and fluid substances composing the organised body and the medium in which it lives, together with the special life-laws of the various tissues constituting the organic structure. ·A hard case:· With celestial phenomena it was much less obvious in what direction the causes were to be looked for (although the relevant causal structures were far simpler than those of society and of life). What happened was this: scientists combined the laws of certain causes and discovered that those laws •explained all the facts that experience had proved concerning the heavenly motions, and •led to predictions that always turned out to be true. It was only then that mankind knew that those were the causes. But whether we can put the question before we can answer it (society, life) or can’t state it until we have become able to answer it (celestial phenomena), either way it must be answered, and the laws of the different causes ascertained, before we can deduce from them the conditions of the effect.

The mode of ascertaining those laws is the fourfold method of experimental inquiry that I already discussed—no other way is even possible. All I need add are a few remarks on the application of that method to cases of the composition of causes.

Obviously, we can’t expect to find the law of a tendency through an induction from cases where the tendency is counteracted. The laws of motion couldn’t have been brought to light from the observation of bodies kept motionless by the equilibrium of opposing forces. Even when the tendency is not •counteracted but merely •modified by having its effects combined with the effects of some other tendency, we are still not well placed to extract from the tangle the law of the tendency itself. It would hardly have been possible to discover the law that every body in motion tends to continue moving in a straight line by an induction from instances in which the motion is deflected into a curve by some external force. In such cases the Method of Concomitant Variations can give some help; but still the principles of a judicious experimentation prescribe that the law of each of the tendencies should be studied, if possible, in cases where that tendency operates •alone or •in combination only with agencies whose effects can be calculated and allowed for.

Accordingly, in cases where the causes can’t be separated and observed apart it’s very hard to lay down with due certainty the inductive foundation needed to support the deductive method. (It’s bad luck that such cases are numerous and important.) This difficulty is especially conspicuous in the case of physiological phenomena, because it’s seldom possible to separate the different agencies that collectively compose an organised body, without destroying the very phenomena we are trying to investigate:

—following life, in creatures we dissect,

We lose it, in the moment we detect.

(Alexander Pope)

For this reason I’m inclined to think that •physiology (greatly and rapidly progressive though it now is) is troubled by greater natural difficulties, and is probably capable of less ultimate perfection, than even •the social sciences. We can make a better job of studying the laws and operations of one human mind apart from other minds than we can of studying the laws of one organ or tissue of the human body apart from the other organs or tissues.

Pathological facts—i.e. in common language, diseases— in their different forms and degrees provide physiologists with the most valuable equivalent to experimentation strictly so-called, because they often show us a definite disturbance in some one organ or organic function, the remaining organs and functions being unaffected, at least for a while. It’s true that. . . .there can’t be a prolonged disturbance in any one function without eventually involving many of the others; and as soon as this happens the experiment loses most of its scientific value. Everything depends on observing the early stages of the disturbance, which unfortunately are bound to be the least conspicuous. But if organs and functions that aren’t disturbed at first become affected in a fixed order of succession, that throws some light on one organ’s influence over another; and we occasionally get a series of effects that we can with some confidence attribute to the original local disturbance; but to get this benefit we have to know that the original disturbance was local. If instead it was ‘constitutional’ (as they say)—i.e. if we don’t know where in the animal’s system it started, or exactly what it consisted of—we can’t determine which of the various upsets was cause and which was effect—which of them were produced by one another, and which by the direct (perhaps delayed) action of the original cause.

. . . .We can also produce pathological facts artificially. We can try experiments, even in the popular sense of the term, by subjecting the living being to some external agent such as the mercury of my former example or cutting a nerve so as to ascertain the functions of different parts of the nervous system. This experimentation isn’t intended to obtain a direct solution of any practical question, but rather to discover general laws from which the conditions of any particular effect can be obtained later by deduction; so it’s best to select cases whose circumstances can be best ascertained; and those are usually not ones in which there’s any practical object in view. The experiments are best tried not in a state of disease, which is essentially a changeable state, but in the condition of health, which is comparatively stable. In sickness, unusual agencies are at work and we can’t predict their results; in health the usual course of the physiological phenomena would remain undisturbed if it weren’t for the disturbing cause that we introduce.

Such are our inductive resources for ascertaining the laws of the causes considered separately when we ·confront them only in complexes and· don’t have any way of separating them out and then investigating them separately. (Actually, the Method of Concomitant Variations sometimes comes to our aid; but it is as burdened as the more elementary methods are by the special difficulties of the subject.

These resources are so glaringly inadequate that the backward state of the science of physiology is no surprise. Indeed our knowledge of causes ·in that science· is so imperfect that it’s no surprise we can’t explain, and couldn’t without specific experience have predicted, many of the facts that we know about from ordinary observation. Fortunately, we’re much better informed than this concerning the empirical laws of the phenomena, i.e. the uniformities that we can’t yet decide how to classify—whether as •cases of causation or merely •results of it. Not only do we have this:

The order in which the facts of organisation and life successively manifest themselves, from the first germ of existence to death, has been found to be uniform and very accurately ascertainable

but also

Variations to the entire facts of comparative anatomy and physiology, the characteristic organic •structure corresponding to each class of •functions has been determined with considerable precision.

We are quite ignorant as to whether these organic conditions are all the conditions—and in many cases we don’t even know whether they are conditions at all rather than mere collateral effects of some common cause. And we’re not likely ever to know this unless we can construct an organised body and try whether it would live.

Those are the obstacles we encounter in cases of this ·complex-cause· sort when we try to take the first step, the inductive step, in applying the Deductive Method to complex phenomena. But fortunately things are usually not as bad as that. In general, the laws of the causes on which the effect depends can be obtained by an induction from comparatively simple instances. . . . By ‘simple instances’ I mean of course ones in which the action of each cause is not much intermixed or interfered with other causes whose laws we don’t know. The use of the Deductive Method to ascertain the laws of a complex effect has sometimes had brilliant results, but only when the induction supplying the premises for the Deductive Method has rested on simple instances of that kind.

§2. Second operation, Ratiocination: When the laws of the causes have been ascertained, and the first stage of our great logical operation satisfactorily completed, the second part follows: determining from the laws of the ·separate· causes what the effect will be of any given combination of those causes. This is a process of ratiocination, and it often involves processes of calculation in the narrow sense in which it = numerical calculation. When our knowledge of the causes is so perfect as to extend to the exact numerical laws that they conform to in producing their effects, the ratiocination may include among its premises the theorems of the science of number—and I’m speaking of the whole immense extent of that science. We often need the most advanced truths of mathematics to be able to compute an effect of which we already know the numerical law; and even with the help of those advanced truths we can’t get very far. Here’s a simple problem:

Given the locations and masses of three bodies that are gravitating toward one another, with a force directly proportional to their mass and inversely proportional to the square of the distance, how do we calculate what their locations will be after n seconds?

All the resources of the calculus haven’t yet been able to provide a general solution—only approximations. [And in 2012 there is still no complete general solution to the ‘three-body problem’.] A slightly more complex case, though still one of the simplest that arise in practice, is that of plotting the motion of a projectile. Even if we know and have numerical values for all the causes that affect the velocity and range of a cannonball—the force of the gunpowder, the angle of elevation, the density of the air, the strength and direction of the wind—it’s an extremely difficult mathematical problem to put these together so as to calculate their combined effect.

Besides the theorems of number, those of geometry also come in as premises, when we are trying to solve problems in mechanics, optics, acoustics, or astronomy—where the effects take place in space and involve motion and extension. But when the complication increases, and the effects depend on so many and such shifting causes that there’s no place for fixed numbers or for straight lines and regular curves, the laws of number and extension are applicable, if at all, only on a large scale where precision of details becomes unimportant. I’m thinking here of physiology, and even more of mental and social phenomena. Although mathematical laws play a conspicuous part in the most striking examples of the investigation of nature by the Deductive Method—e.g. in the Newtonian theory of the celestial motions—they are by no means an indispensable part of every such process. All that is essential is reasoning from a general law to a particular case—i.e. determining by means of the particular circumstances of that case what result is required in that instance to fulfill the law. Thus in the Torricellian experiment [in effect, the discovery of the barometer], if the fact that air has weight had been previously known it would have been easy, without any numerical data, to deduce from the general law of equilibrium that the mercury would stand in the tube at a height such that the column of mercury would exactly balance a column of the atmosphere of equal diameter. . . .

By such ratiocinations from the separate laws of the causes we can to some extent answer either of the following questions:

•Given a certain combination of causes, what effect will follow?

•What combination of causes, if it existed, would produce a given effect?

In the one case, we determine the effect to be expected in any complex circumstances of which the different elements are known; in the other case we learn according to what law—under what antecedent conditions—a given complex effect will occur.

§3. Third operation, Verification: You may want to say:

Those arguments that you used to dismiss as illusory the methods of direct observation and experiment when applied to the laws of complex phenomena— don’t they apply with equal force against the Method of Deduction? When in every single instance a multitude (often an unknown multitude) of agencies are clashing and combining, how can we be sure that in our a priori computation we have taken them all into account? Aren’t there certain to be many that we don’t know anything about? Aren’t we likely to have overlooked some that we do know of? And even if we did take account of them all, that would be useless unless we knew the precise numerical law of each, which we usually don’t. And if we did, we would need to make a calculation which, in any but very simple cases, surpasses the utmost power of mathematical science with all its most modern improvements.

These objections have real weight, and would be unanswerable if it weren’t for the fact that when we are using the Deductive Method there’s a test that enables us to judge whether we have committed any of those errors. The application of this test constitutes Verification, the third essential component part of the Deductive Method, without which all the results the method can give amount to little more than conjectures. We aren’t entitled to rely on the general conclusions arrived at by deduction unless careful comparison shows us that they fit the results of direct observation wherever that can be had. If we have relevant experience and it confirms them, we may safely trust to them in other cases of which we don’t yet have specific experience. But if our deductions have led to the conclusion that from a particular combination of causes C a given effect E would result, then in all known cases where C is not followed by E we must be able to show (or at least to make a probable surmise about) what blocked E; and if we can’t do that the theory is imperfect and not yet to be relied upon. And the verification isn’t complete unless some of the cases where the theory is confirmed by the observed result are at least as complex as any where its application could be called for.

If direct observation and the assembling of instances have provided us with any relevant empirical laws (whether true in all observed cases, or only true for the most part), the best verification the theory could have would be its leading deductively to those empirical laws, so that the complete or incomplete uniformities that were observed among the phenomena were accounted for by the laws of the causes. . . . It was very reasonably thought to be an essential requirement of any true theory of the causes of the celestial motions that it should lead by deduction to Kepler’s laws—which the Newtonian theory did.

Something else that is important for the verification of theories obtained by deduction. . . .is that the phenomena should be described in the most comprehensive and accurate manner possible;. . . .as when the series of the observed places of a planet was first expressed by a circle, then by a system of epicycles, and subsequently by an ellipse.

Complex instances would have been no use for the discovery of the simple laws into which we ultimately analyse their phenomena, but when they have served to verify the analysis they become additional evidence for the laws themselves. Although we couldn’t have discovered the law from complex cases, still when the law—discovered in some other way—is found to be in accordance with the result of a complex case, that case becomes a new experiment on the law, and helps to confirm something that it didn’t help to discover. . . . This was strikingly conspicuous in the example [here] in which the difference between the observed and the calculated velocity of sound was found to result from the heat developed by the condensation that happens in each sound-vibration. This was a trial, in new circumstances, of the law of the development of heat by compression, and it added materially to the proof of the universality of that law. Accordingly, any law of nature is regarded as having become more certain by being found to explain some complex case that hadn’t previously been thought of in connection with it; and this indeed is a consideration that scientific inquirers customarily value too much rather than too little.

To the Deductive Method—with its three constituent parts, Induction, Ratiocination, and Verification—the human mind is indebted for its most conspicuous triumphs in the investigation of nature. To it we owe all the theories by which vast and complicated phenomena are embraced under a few simple laws that could never have been detected by the direct study of those great phenomena. To get a sense of what the method has done for us, consider the case of planetary motions. Of the greater instances of the composition of causes this is one of the simplest, because (except in a few not very important cases) each heavenly body x can be considered (without material inaccuracy) to be never attracted by more than two bodies at once, •the sun and •one other planet or satellite. So x’s motions depend on only four different agents:

•the sun,

•the other planet or satellite,

•the reaction of x itself, and

•the force generated by x’s own motion and acting in the direction of the tangent.

This is surely a much smaller number than any of the other great phenomena of nature is determined or modified by. Yet how could we ever have discovered the combination of forces on which the motions of the earth and planets depend by merely comparing the orbits or velocities of different planets, or the different velocities or positions of the same planet ·at different times·? Despite the usual regularity of those motions, and although the periodical recurrence of exactly the same effect shows that all the combinations of causes that occur at all recur periodically, we wouldn’t have known what the causes were if the existence of precisely similar agencies on our own earth hadn’t brought the causes themselves within the reach of experimentation under simple circumstances. I’ll have occasion in chapter 14 to analyse this great example of the method of deduction, so I shan’t spend time on it here. My next topic is a secondary application of the deductive method, the result of which is not to •prove laws of phenomena but to •explain them.

Chapter 12. Explaining laws of nature

§1. When we use the deductive operation to derive the law of an effect from the laws of the causes that jointly give rise to it, we may be engaged in either of two things: •discovering the law or •explaining a law already discovered. The word ‘explanation’ occurs so continually in philosophy, and has such an important place in it, that a little time spent in fixing its meaning will be well spent.

An individual fact is said to be explained when someone points out its cause, i.e. states the law or laws of causation of which its production is an instance. A fire is explained when it is proved to have arisen from a spark falling onto a heap of dry leaves. Similarly a law or uniformity in nature L is said to be explained when someone points out another law or laws •of which L is a special case and •from which it could be deduced.

§2. There are three sets of circumstances in which a law of causation can be explained from other laws—or, as it also is often expressed, resolved into other laws.

The first is a case that I have already fully considered: a mixture of laws producing a joint effect equal to the sum of the effects of the causes taken separately. The law of the complex effect is explained by being resolved into the separate laws of the causes that contribute to it. For example, •the law of the motion of a planet is resolved into •the law of the acquired force that tends to produce a uniform motion in the tangent and •the law of the centripetal force that tends to produce an accelerating motion toward the sun, the real motion being a compound of those two.

In this resolution of the law L of a complex effect, the laws L is compounded of aren’t the only elements. It is resolved into the laws of the separate causes and the fact of their coexistence. This is as essential as the other ingredients, whether we are discovering L or only explaining it. To deduce the laws of planetary motions, we have to know not only

•the law of a rectilineal force and

•the law of gravitative force, but also

•the fact that both these forces exist in the celestial regions, and even their relative amount.

The complex laws of causation are thus resolved into two distinct kinds of elements: •simpler laws of causation and •collocations, the latter consisting in the existence of certain agents or powers in certain places at certain times. Later on I’ll need to return to this distinction and discuss it at some length—enough to remove any need to go on about it here. So: the first kind of explanation of laws of causation occurs when the law of an effect is resolved into the various tendencies of which it is the result, together with the laws of those tendencies. . . .

§3. A second kind of explanation of laws occurs when what seemed to be an immediate cause-effect pair turns out to have an intermediate link, a fact caused by the antecedent and in its turn causing the consequent. A seemed to be the ·immediate· cause of C, but we later found that A was the ·immediate· cause only of B and a remote cause of C, and that B was the ·immediate· cause of C. We knew that touching an outward object caused a sensation. We discovered later that •after we have touched the object and •before we experience the sensation •some change occurs in a kind of thread called a ‘nerve’ that extends from our external organs to the brain. Thus, touching the object is only the remote cause of our sensation—i.e. not the cause properly speaking, but the cause of the cause—and the real cause of the sensation is the change in the nerve. Future experience may not only •increase our knowledge of the nature of this change, but also •interpolate another link. It may be (for example) that between the contact of the object with our external organs and the change in the nerve there is some electric phenomenon, or some phenomenon unlike anything we now know. No such intermediate link has been discovered up to now, so the touch of the object must be regarded provisionally as the immediate cause of the event in the nerve. Thus the sequence

contact with an object → sensation of touch

is discovered not to be an ultimate law; it is ‘resolved’ (as they say) into two other laws:

contact with an object → event in the nerve, and event in the nerve → sensation of touch.

Another example: the more powerful acids corrode or blacken organic compounds; this is causation, but ·only· remote causation; and it’s said to be explained when it is shown that there’s an intermediate link, namely an event in which chemical elements in the organic structure separate from the rest and combine with the acid. The acid causes this separation of the elements, and the separation of the elements causes the disorganisation and often the charring of the structure. . . .

§4. This is important: when a sequence of phenomena is thus resolved into other laws, they’re always laws more general than itself. The law that A is followed by C is less general than either of the laws connecting B with C and A with B. Some very simple points will show that this is so.

All laws of causation are liable to be counteracted or frustrated by the non-fulfillment of some negative condition; so B’s tendency to produce C may be defeated. Now the law that A produces B is equally fulfilled whether or not B is followed by C; but the law that •A produces C by means of B is of course fulfilled only when B really is followed by C; so it is less general than the law that •A produces B. It is also less general than the law that B produces C. [Mill’s defence of this is the same, mutatis mutandis, as the one he has just given. And then he has a paragraph applying all this to the touch-nerve-sensation case, including this:] The law that an event in a nerve produces sensation is more general than the law that contact with an object produces sensation, because the sensation equally follows the change in the nerve when it is produced not by contact with an object but by some other cause. . . .

The laws of •more immediate sequence that the •law of a remote sequence is resolved into are not only more general than that law is but also more to be relied on. . . . The tendency of A to produce C can be defeated by whatever can defeat either the tendency of A to produce B or the tendency of B to produce C; so it is twice as liable to failure as either of those more elementary tendencies; and the generalisation that A is always followed by C is twice as likely to be found erroneous. . . .

The resolution of one generalisation into two others not only •shows that there are possible failures of the former from which its two elements are exempt, but also •shows where these are to be looked for. As soon as we know that B intervenes between A and C we also know that if there are cases where A → C doesn’t hold they are most likely to be found by studying the causes and the effects of B.

So we see that in the second of the three ways in which a law can be resolved into other laws, the explaining laws are more general (i.e. cover more cases) and less likely to collide with subsequent experience than is the law they explain. They are

•more nearly unconditional,

•defeated by fewer contingencies, and

•a nearer approach to universal truths of nature.

And all this is still more obviously true of first of the three modes of resolution. When the law of an effect of combined forces is resolved into the separate laws of the causes, the law of the effect must be less general than the law of any of the causes because it only holds when they are combined; whereas the law of any one of the causes holds good both in that combination and out of it. . . .

Here’s another strong reason why the law of a complex effect must be less general than the laws of the causes that collaborate to produce it. If two complexes involve the same causes acting according to the same laws, they can still differ in the proportions in which the causes are combined; and that can lead to their having effects that differ not merely in •quantity but in •kind. The combination of •a centripetal force with •a projectile force, in the proportions they have in all the planets and satellites of our solar system, gives rise to an elliptical motion; but if the ratio between the two forces were slightly different the motion they produced would be in a circle or parabola or hyperbola. . . . The law of each of the concurrent causes remains the same, however their collocations may vary; but the law of their joint effect varies with every difference in the collocations. . . .

§5. There is also a third mode in which laws are resolved into one another; and in this it’s self-evident that they are resolved into laws more general than themselves. This third mode is the subsuming (as they say) of one law under another, i.e. the gathering up of several laws into one more general law that includes them all. The most splendid example of this occurred when •terrestrial gravity and •the central force of the solar system were brought together under •the general law of gravitation. It had already been proved that the earth and the other planets tend toward the sun; and it had been known from the earliest times that terrestrial bodies tend toward the earth. These were similar phenomena; for them both to be subsumed under one law all that was needed was to prove that as well as being alike in •quality they conform to the same rules as to •quantity. This was first shown to be true of the moon: it resembled terrestrial objects in tending to a centre and indeed in tending toward the earth. After it had been discovered that the moon’s tendency toward the earth varied inversely with the square of the distance between them, it was directly calculated that if the moon were as near to the earth as terrestrial objects are, and if the acquired force in the direction of the tangent were suspended, the moon would fall toward the earth through exactly as many feet per second as those objects do by virtue of their weight. So the inference was irresistible •that the moon also tends toward the earth by virtue of its weight, and •that these two phenomena. . . .are cases of one and the same law of causation. But the tendency of the moon to the earth, and the tendency of the earth and planets to the sun, were already known to be cases of the same law of causation; and thus the law of all these tendencies and the law of terrestrial gravity were recognised as identical, and were subsumed under one general law, that of gravitation.

In a similar manner the laws of magnetic phenomena have more recently been subsumed under known laws of electricity. That’s how the most general laws of nature are usually arrived at—we climb up to them by successive steps. Here is why. To arrive by correct induction at laws holding under such an immense variety of circumstances, laws so general as to be independent of any changes of space or time that we can see, requires many sets of experiments or observations, conducted at different times by different people. One set of observations teaches us that the law holds good under conditions C_1, another that it holds good under conditions C_2, and by combining these we find that it holds good under much more general conditions or even holds universally. The general law is literally the sum of all the partial ones: it recognises the same sequence in different sets of instances, and can in fact be regarded as merely one step in the process of elimination. The tendency of bodies toward one another that we now call ‘gravity’ was at first observed only on the earth’s surface, where it shows up only as a tendency of all bodies toward the earth; it might have been regarded as a special property of the earth itself, because one of the circumstances—namely proximity to the earth—hadn’t been eliminated. To eliminate this required a fresh set of instances in other parts of the universe; we couldn’t create these ourselves, and though nature had created them for us, we weren’t well-placed to observe them. The making of these observations came within the province of a different set of scientists from those who studied terrestrial phenomena; and it was a matter of great interest back at a time when the idea of explaining •celestial facts by •terrestrial laws was regarded as the confounding of an indefeasible distinction [= ‘the crossing of an uncrossable line’]. But when the celestial motions were accurately ascertained and the deductions performed, showing that their laws corresponded with the laws of terrestrial gravity, those celestial observations became a set of instances that precisely eliminated the circumstance of proximity to the earth. This proved that in the case of terrestrial objects the cause of the downward motion or pressure was not

•the earth as such, but

•the presence of some great body within certain limits of distance,

this being the circumstance common to the terrestrial and the celestial instances.

§6. There are, then, three ways of explaining laws of causation, i.e. resolving them into other laws. (1) The law of an effect of combined causes is resolved into the separate laws of the causes together with the fact of their combination. (2) The law that connects two links (not immediate neighbours) in a chain of causation is resolved into the laws that connect each with the intermediate links. (3) After a law has been shown to hold good in several classes of cases we decide that what is true in each of these is true under some more general supposition consisting of what all those classes of cases have in common. The first two involve resolving one law into two or more; the third resolves two or more into one. . . .

In all three processes, laws are resolved into laws more general than themselves—laws extending to all the cases that the former extended to, and others besides. In the first two they are also resolved into laws that are more certain—i.e. more universally true—than themselves. They aren’t proved to be •laws of nature, because that would require them to be universally true; what they are proved to be is •results of laws of nature. With that status, they’re only conditionally true, usually true. Not so with the third process, because here the partial laws are in fact the very same law as the general one, so any exception to them would be an exception to it too. . . .

I’m using ‘explanation’ in its philosophical sense. Explaining one law of nature by another is merely substituting one mystery for another, and does nothing to make the general course of nature other than mysterious. We can no more assign a why for the more extensive laws than for the partial ones. In ordinary talk about these matters, an ‘explanation’ replaces a mystery that is still strange by one that has become familiar and come to seem not mysterious. But the process I have been discussing here often does the exact opposite: it resolves •a phenomenon that we are familiar with into •one of which we previously knew little or nothing; as when the common fact that heavy bodies fall was resolved into the tendency of all particles of matter toward one another. Don’t forget this: in science those who speak of ‘explaining’ a phenomenon mean (or should mean) to be pointing out not some more familiar phenomenon but merely

•some more general phenomenon of which it is a partial exemplification, or

•some laws of causation that produce it by their combined action, and from which its conditions can therefore be determined deductively.

Every such operation brings us a step nearer toward answering the question that I said here includes the whole problem of the investigation of nature, namely: What are the fewest and simplest assumptions, which being granted the whole existing order of nature would result? What are the fewest general propositions from which all the uniformities existing in nature could be deduced?

. . . .In minds that aren’t used to accurate thinking there is often a confused notion that the general laws are the causes of the partial ones, e.g. that the law of general gravitation causes the phenomenon of the fall of bodies to the earth. But that’s a misuse of the word ‘cause’; terrestrial gravity isn’t an •effect of general gravitation but a •case of it, i.e. one kind of the particular instances in which that general law obtains. . . .

Chapter 13. Examples of the explanation of laws of nature

§1. The most striking example of the kind of explanation I have been talking about—explaining causal laws and regularities among •special phenomena by resolving them into laws that are simpler and more •general—is the great Newtonian generalisation. So much has already been said about this ·that you don’t need me to expound it to you·; it’s enough to call attention to the great number and variety of the special observed uniformities that it accounts for as particular cases or consequences of one very simple law of universal nature. The simple fact that

every particle of matter tends toward every other particle, with the tendency varying inversely as the square of the distance between them

explains •the fall of bodies to the earth, •the revolutions of the planets and satellites, •the motions (as far as we know them) of comets, and •all the regularities that have been observed in these special phenomena, such as the

•elliptical orbits, and the variations from exact ellipses,

•the relation of the planets’ distances from the sun to the duration of their revolutions,

•the precession of the equinoxes [see Glossary],

•the tides,

and a vast number of minor astronomical truths.

[Mill reminds us of the explanation of magnetism in terms of electricity; and mentions the explanation—not yet complete, but already powerful—of the properties of the bodily organs in terms of the elementary properties of the tissues making them up.]

Another striking instance is Dalton’s generalisation, commonly known as the ‘atomic theory’. It had been known from the start of detailed chemical observation that any two bodies combine chemically with one another in only a certain number of proportions; but those proportions were always expressed in percentages by weight. . . .; and those formulations didn’t let the chemists see any relation between •the proportion in which a given element combines with one substance and •the proportion in which it combines with others. Dalton’s great step consisted in perceiving that a unit of weight—known now as ‘the atomic weight’—might be established for each substance, such that by supposing the substance to enter into all its combinations in the ratio of that unit (or of some low multiple of it) all the different proportions that had previously been expressed by percentages would result. Thus taking 1 to be the atomic weight of hydrogen and 8 to be the atomic weight of oxygen,

•the combination of one unit of hydrogen with one unit of oxygen would produce the exact proportion (by weight) between the two substances that is known to exist in water;

•the combination of one unit of hydrogen with two units of oxygen would produce the proportion that exists in the other compound of those two elements, hydrogen peroxide; and

•and the combinations of hydrogen and of oxygen with all other substances would fit the supposition that •those two enter into combination by single units, or twos, or threes of their atomic weight = 1 and 8, and •the other substances enter the combinations by ones or twos or threes of their atomic weights.

The result is that a table of the atomic weights of all the elementary substances comprises in itself, and scientifically explains, all the proportions in which any substance, elementary or compound, can enter into chemical combination with any other substance whatever.

§2. [Mill praises the work of Thomas Graham, who highlighted the difference between the crystalloid and colloidal states of matter, and discovered many of their properties. Crucially:] Whereas colloidal substances are easily penetrated by water and by the solutions of crystalloid substances, they are very little penetrable by one another. That enabled Graham to introduce a highly effective process (called ‘dialysis’) for separating the crystalloid substances contained in any liquid mixture, by passing them through a thin wall of colloidal matter that allows through little if any colloidal material. This enabled Graham to account for a number of special results of observation that hadn’t previously been explained. [Mill sketches three of them and then this fourth:] Much light is thrown on the observed phenomena of osmosis (the passage of fluids outward and inward through animal membranes) by the fact that the membranes are colloidal. The result of that is that the water and saline solutions contained in the animal body pass easily and rapidly through •the membranes, whereas the substances directly applicable to nutrition, which are mostly colloidal, are detained by •them.

Salt’s property of preserving animal substances from putrefaction is resolved by Liebig into two more general laws: •salt’s strong attraction for water, and •the need for water if putrefaction is to occur. The intermediate item interpolated here between the remote cause and the effect isn’t merely inferred but can be seen; for we’ve all seen that flesh on which salt has been thrown is soon swimming in brine.

The need of water for putrefaction itself provides an additional example of the explanation of laws. The law itself is proved by the Method of Difference: flesh completely dried and kept in a dry atmosphere doesn’t putrefy. . . . A deductive explanation of this same law results from Liebig’s speculations. The putrefaction of animal and other nitrogencontaining bodies is a process in which they are gradually converted into (mainly) carbonic acid and ammonia. Now,

•to convert the carbon of the animal substance into carbonic acid requires oxygen, and

•to convert the nitrogen into ammonia requires hydrogen,

and these two are the elements of water. . . .

§3. Among the many important properties of the nervous system that were first discovered or strikingly illustrated by Brown-Séquard, I select the reflex influence of the nervous system on nutrition and secretion. By reflex nervous action is meant

action that one part of the nervous system exerts over another part ·of the body· independently of the will and probably without passing through the brain and thus without consciousness.

Many experiments have shown that irritation of a nerve in one part of the body can in this way start up powerful action in another part; for example,

•food injected into the stomach through a divided oesophagus ·and by-passing the tongue· nevertheless produces secretion of saliva;

•warm water injected into the bowels, and various other irritations of the lower intestines, excite secretion of the gastric juice,

and so on. The reality of the power being thus proved, its agency explains a great variety of apparently anomalous phenomena, of which I select the following from Brown-Séquard’s Lectures on the Nervous System:

•The production of tears by irritation of the eye, or of the mucous membrane of the nose;

•The secretions of the eye and nose increased by exposure of other parts of the body to cold;

•Inflammation of one eye, especially when cause by trauma, often excites a similar state in the other eye, which can be cured by cutting the intervening nerve;

•Loss of sight is sometimes produced by neuralgia, and has been known to be immediately cured by the extraction of a diseased tooth;

•A cataract has been produced in a healthy eye by a cataract in the other eye, or by neuralgia, or by a wound of the frontal nerve;

•The well-known phenomenon of a sudden stoppage of the heart’s action, and consequent death, produced by irritation of some of the nerve-ends—e.g. by drinking very cold water, or by a blow on the abdomen or other sudden excitation of the abdominal sympathetic nerve, though this nerve can be irritated to any extent without stopping the heart’s action if the nerves connecting them are cut;

•An extensive burn on the surface of the body can produce extraordinary effects on the internal organs—violent inflammation of the tissues of the abdomen, chest, or head; when death ensues from this kind of burn this internal disturbance is one of the most frequent causes of it;

•Paralysis and anaesthesia of one part of the body from neuralgia in another part; and muscular atrophy from neuralgia, even when there is no paralysis;

•Tetanus produced by cutting a nerve. Brown-Séquard thinks it highly probable that hydrophobia is a phenomenon of a similar nature;

•changes in the nutrition of the brain and spinal cord, manifesting themselves by epilepsy, chorea, hysteria, and other diseases, occasioned by lesion of some of the nerve-ends, e.g. worms, stones, tumours, diseased bones, and in some cases even by slight irritations of the skin.

§4. From these and similar instances we can see that when a previously unknown law of nature is brought to light or when new light has been thrown on a known law by experiment, it’s important to examine all cases that include the conditions needed to bring that law into action. This process leads to demonstrations of •previously unsuspected special laws and of •explanations of laws that are already empirically known.

For example, Faraday discovered by experiment that if a conducting body is set in motion at right angles to the direction of a natural magnet, voltaic electricity is generated; and he found this to hold not only of small magnets but of that great magnet the earth. With that law established experimentally, we can now watch out for fresh instances in which a conductor moves or revolves at right angles to the direction of the earth’s magnetic poles. In each of these we can expect electricity to be generated. In the northern regions where the polar direction is nearly perpendicular to the horizon, all horizontal motions of conductors will produce electricity: horizontal wheels made of metal, for example, and all running streams will generate an electric current that circulates round them; and the air thus charged with electricity may be one cause of the Aurora Borealis. In the equatorial regions, on the other hand, upright wheels placed parallel to the equator will create a voltaic circuit, and waterfalls will naturally become electric.

For a second example, it has been proved (mainly by Graham’s researches) that gases have a strong tendency to permeate animal membranes, and diffuse themselves through the spaces that such membranes enclose, even if there are already other gases in those spaces. [Mill uses this to ‘demonstrate or explain’ six ‘more special laws’.]

§5. . . . .There are countless examples of new theories agreeing with and explaining old empiricisms [Mill’s phrase]. All the sound things that experienced persons have said about human character and conduct are simply special laws that the general laws of the human mind explain and resolve. The empirical generalisations on which the operations of the arts [see Glossary] have usually been based are continually •justified and confirmed on the one hand, or •corrected and improved on the other, by the discovery of the simpler scientific laws that those operations depend on for their effectiveness. The effects of the rotation of crops, of the various manures, and other processes of improved agriculture, have been for the first time resolved in our own day into known laws of chemical and organic action, by Davy, Liebig, and others. The processes of the medical art are even now mostly empirical; the effectiveness of each is inferred from a special and most precarious experimental generalisation; but as science advances in discovering the simple laws of chemistry and physiology, progress is made in discovering •the intermediate links in the series of phenomena and •the more general laws they depend on; and thus, while the old processes are either exploded or. . . .explained, better processes, based on the knowledge of immediate causes, are continually being suggested and brought into use. [In a footnote Mill gives an example of a surgical improvement born of an explanation of the partial success of an ‘old’ procedure.] Many of the truths of geometry, even, were generalisations from experience before they were deduced from first principles. The quadrature of the cycloid [see Glossary] is said to have been first effected by weighing a cycloidal card and comparing its weight with that of a piece of similar card of known dimensions.

§6. To the foregoing examples from physical science I’ll add another from mental science. Here is one of the simple laws of mind:

Ideas of a pleasurable or painful sort form associations more easily and strongly than other ideas, i.e. they become associated ·with other ideas· after fewer repetitions, and the association is more durable.

This is an experimental law based on the Method of Difference. It is possible by deduction from this law to demonstrate and explain many of the more special laws that experience shows to exist among particular mental phenomena, for example:

•how fast and easily thoughts connected with our passions or our more cherished interests are aroused, and how durably they stick in our memory;

•the vivid recollection we retain of tiny circumstances that accompanied any object or event that deeply interested us, and of the times and places in which we have been very happy or very miserable;

•the horror with which we view the accidental instrument of any occurrence that shocked us or the place where it happened, and the pleasure we get from any reminder of past enjoyment;

all these effects being proportional to the sensibility of the individual mind, and to the consequent intensity of the pain or pleasure from which the association originated. James Martineau has suggested that this same elementary law of our mental constitution, suitably followed out, would explain a variety of previously unexplained mental phenomena, and in particular some of the basic differences among human characters and mental abilities. Associations are of two sorts—between synchronous impressions and between successive impressions—and the law that makes associations stronger in proportion to the pleasurable or painful character of the impressions operates with special force in the synchronous class of associations. Martineau remarks that in minds with a strong organic sensibility, synchronous associations are likely to predominate, producing a tendency to conceive things in pictures and in concrete detail, richly clothed in attributes and circumstances, a mental habit that is commonly called ‘imagination’ and is one of the special qualities of the painter and the poet; while persons who are more moderately susceptible to pleasure and pain will tend to associate facts chiefly in the order of their succession; and such persons, if they have good intellects, will devote themselves to history or science rather than to creative art. I have tried (in my Dissertations and Discussions, vol. 1, fourth paper) to pursue this interesting speculation further, and to examine what help it can give towards explaining the poetical temperament. It at least serves as an example to show what scope there is for deductive investigation in the important and hitherto so imperfect science of mind.

§7. I have presented many examples of the discovery and explanation of special laws of phenomena by deduction from simpler and more general ones, because I wanted to characterise the Deductive Method clearly and give it the prominence its importance deserves. The Deductive Method is destined from now on to predominate in the course of scientific investigation. A revolution is peaceably and progressively going on in philosophy [here = ‘the philosophy of science’], the reverse of the revolution to which Bacon attached his name. That great man changed the method of the sciences from •deductive to •experimental, and it is now rapidly reverting from •experimental to •deductive. But the deductions that Bacon abolished were from premises hastily snatched up or arbitrarily assumed. The axioms weren’t established by legitimate canons of experimental inquiry, nor were the results tested by that indispensable element of a rational Deductive Method, verification by specific experience. Between the primitive method of deduction ·that Bacon opposed· and the one I have tried to characterise there is all the difference that separates Aristotelian physics from the Newtonian theory of the heavens.

But don’t think that all—or even most—of the great generalisations from which the subordinate truths of the more backward sciences will some day be deduced by reasoning. . . .are truths that are now known and accepted. We can be sure that many of the most general laws of nature are as yet entirely unthought-of; and that many others that will eventually qualify as general laws of nature are now known only as laws or properties of some limited class of phenomena. (Just as electricity, now recognised as one of the most universal of natural agencies, was once known only as an odd property that certain substances acquired by friction, of first attracting and then repelling light bodies.) If the theories of heat, cohesion, crystallisation, and chemical action are destined—as surely they are—to become deductive, the truths that will then be regarded as the premises of those sciences would probably strike us now as being as novel as the law of gravitation appeared to Newton’s contemporaries; perhaps even more novel than that. because Newton’s law was an extension of the law of weight—i.e. of a generalisation familiar from of old and already covering a considerable body of natural phenomena. The general laws of a similarly commanding kind that we still look forward to the discovery of may not always find so much of their foundations already laid!

These general truths will doubtless make their first appearance as hypotheses; not proved or even provable at first but assumed as premises for the purpose of deducing from them the known laws of concrete phenomena. But this initial state can’t be what they end up with. To entitle an hypothesis to be accepted as a truth of nature, and not as a mere technical help to the human faculties, it must be testable by the canons of legitimate induction, and must actually have been submitted to that test. When this is done successfully, premises will have been obtained from which all the other propositions of the science will from then on be presented as conclusions, and the science will by means of a new and unexpected induction be made deductive.

Chapter 14. The limits to the explanation of laws of nature. Hypotheses

§1. We have been led to recognise a distinction between two kinds of laws or observed uniformities in nature: •ultimate laws and what we could call •derivative laws. Derivative laws are those that can be deduced from other and more general ones, and can indeed be resolved into them. Ultimate laws are those that can’t. We aren’t sure that any of the uniformities we’re now acquainted with are ultimate laws; but we know that there must be ultimate laws, and that every resolution of a derivative law into more general laws brings us nearer to them.

Since we are continually discovering that

•uniformities we thought were ultimate are only derivative, and resolvable into more general laws,

or to put the same thing in different words, since we are continually discovering

•the explanation of some sequence that we previously knew only as a ·brute unexplained· fact,

the question arises: Are there any necessary limits to this philosophical operation, or might it proceed until all the uniform sequences in nature are resolved into some one universal law? This does seem at first sight to be where the progress of induction by the Deductive Method. . . .is heading. Projects of this kind were universal in the infancy of philosophy; any theoretical ideas that held out a less brilliant prospect were regarded in those early times as not worth pursuing. And the idea seems plausible in the light of the remarkable achievements of modern science, so that even now theorists frequently turn up either claiming to have solved the problem or suggesting ways in which it may one day be solved. Even when such large claims aren’t being made, the nature of the solutions that are given or sought for particular classes of phenomena often involves conceptions of what explanation is that would make the notion of explaining all phenomena by means of one cause or law perfectly admissible.

§2. So it’s useful to remark that the number of laws of nature can’t possibly be smaller than the number of distinguishable sensations or other feelings of our nature—I mean distinguishable from one another in quality and not merely in quantity or degree. For example: there’s a phenomenon sui generis [see Glossary] called colour that our consciousness tells us isn’t a particular degree of some other phenomenon such as heat or odour or motion, but intrinsically unlike all others; and it follows from this that there are ultimate laws of colour—that though it may be possible to explain the facts of colour, they can never can be explained from laws of heat or odour alone, or of motion alone, but that however far the explanation goes it will always contain a law of colour. I’m not denying this:

It might be shown that some other phenomenon— e.g. some chemical or mechanical action—invariably precedes and causes each phenomenon of colour.

If this were proved, it would be an important extension of our knowledge of nature; but it wouldn’t explain how or why a motion or chemical action produces a sensation of colour. However hard we studied the phenomena, and however many hidden links we detected in the chain of causation terminating in the colour, the last link would still be a law of colour and not a law of motion or of any other phenomenon. This applies not only to colour as compared with any other of the great classes of sensations, but also to each particular colour as compared with other colours. White colour can’t possibly be explained exclusively by the laws of the production of red colour! In any attempt to explain it, we can’t help including as one element of the explanation the proposition that some antecedent or other produces the sensation of white.

So the ideal limit of the explanation of natural phenomena (towards which we are constantly tending, while knowing that we can’t ever completely attain it) would be to show that each distinguishable variety of our sensations or other states of consciousness has only one sort of cause; e.g. that there is some one condition or set of conditions that is always present whenever we perceive a white colour, and that always produces that sensation in us. As long as there are several known modes of production of a phenomenon (e.g. several substances that have the property of whiteness but no other resemblance that we can find) it’s always possible that one of these modes of production is resolved into another, or that all of them are resolved into some more general newly discovered mode of production. But when the modes of production are reduced to •one, we can’t simplify things any further. This •one may not after all be the ultimate mode; there may be other links to be discovered between the supposed cause and the effect; but the only way we can we can resolve the known law is by introducing some other law that wasn’t previously known, which won’t reduce the number of ultimate laws.

In what cases has science been most successful in explaining phenomena by resolving their complex laws into laws of greater simplicity and generality? [Mill answers that the greatest success is with ‘mechanical motion’, and says that that’s what might be expected: Motion occurs everywhere, it is produced in countless different ways, and the differences between different instances of motion don’t bring in anything that looks like an uncrossable line, like that between colour and odour. He continues:] So there’s no absurdity in supposing that all motion may be produced in one way, by the same kind of cause. And the greatest achievements in physical science have consisted in resolving one observed law of the production of motion into the laws of other known modes of production, or the laws of several such modes into one more general mode; as when

•the fall of bodies to the earth and the motions of the planets were brought under the one law of the mutual attraction of all particles of matter;

•when the motions said to be produced by magnetism were shown to be produced by electricity;

•when the motions of fluids in a lateral direction, or even contrary to the direction of gravity, were shown to be produced by gravity;

and so on. There are many causes of motion that aren’t yet resolved into one another—gravitation, heat, electricity, chemical action, nervous action, and so on—but the attempt of the present generation of scientists to resolve all these different modes of production into one is perfectly legitimate, whether or not it ultimately succeeds. . . .

I needn’t extend this illustration to other cases—the propagation through space of light, sound, heat, electricity, etc. or any of the other phenomena that have been explained by resolving their observed laws into more general laws. I have said enough to display the difference between •the kind of ‘explanation’ and ‘resolution’ of laws that is chimerical and •the kind that it’s science’s great aim to accomplish; and to show into what sort of elements the resolution must be effected, if at all.[1]

§3. From opposing the view that there is only one ultimate law of nature I now turn to the view that there are enormously many of them. (Almost every principle of a true method of doing science needs to be guarded against errors on both sides!) [Comte committed the latter error, Mill writes. His account of how is hard to follow, and his reply to it is omitted here.]

The really weak point in the attempts that have been made to account for colours by the vibrations of a fluid is not that the attempt itself is unscientific but that the existence of the fluid, and the thesis that it vibrates, are simply assumed, purely because they are supposed to help with the explanation of the phenomena. This leads to the important question of the proper use of scientific hypotheses. You don’t need me to explain the connection between •this topic and •the topic of the explanation of natural phenomena and of the unavoidable limits to that explanation.

§4. An hypothesis is a supposition that we make (on admittedly insufficient evidence, or on none) in an attempt to deduce from it conclusions that conform to facts that we know to be real [= ‘to factual propositions that we know to be true’]. The idea is that if •the hypothesis leads to known truths then •it either must be—or at least is likely to be—true. If the hypothesis concerns the cause or mode of production of a phenomenon, it will serve (if accepted) to explain any facts that can be deduced from it; and that is the purpose behind many hypotheses, perhaps most of them. To ‘explain’ something—in the scientific sense of the word—is to resolve [see Glossary]

•a uniformity that isn’t a law of causation into the laws of causation from which it follows, or

•a complex law of causation into simpler and more general ones from which it follows.

If we don’t know any laws that fulfill this requirement, we can invent or imagine some that would fulfill it; and this is making an hypothesis.

Because a hypothesis is a mere supposition, the only limits to hypotheses are the limits of the human imagination; if we want to, we can offer to explain some effect by imagining some cause of an utterly unknown kind acting according to a perfectly fictitious law. But hypotheses of this sort •wouldn’t have any of the plausibility of the ones that ally themselves by analogy with known laws of nature, and wouldn’t meet the desire that made-up hypotheses are generally invented to satisfy, namely enabling the imagination to represent to itself an obscure phenomenon in a familiar light. So there has probably been no hypothesis in the history of science in which both the agent and the law of its operation were fictitious. ·In every actual hypothesis·, either (i) the supposed the cause is real but the law according to which it acts is merely supposed, or (ii) the cause is fictitious but the laws it is supposed to operate by are similar to the laws of some known class of phenomena. An instance of (i) is provided by the different suppositions made regarding the law of the planetary central force before the true law was discovered. That law, namely that the force varies as the inverse square of the distance, first suggested itself to Newton as an hypothesis, and was verified by proving that it led deductively to Kepler’s laws. Hypotheses of kind (ii) include

•the vortices [see Glossary] of Descartes, which were fictitious but were supposed to obey the known laws of rotatory motion; and

•the two rival hypotheses regarding the nature of light, one ascribing the phenomena to a fluid emitted from all luminous bodies, the other (now generally received) attributing them to vibratory motions among the particles of an ether—·a super-thin fluid·—pervading all of space.

Of the existence of either fluid there is no evidence except the explanation they offer for some of the phenomena; but they’re supposed to produce their effects according to known laws—in one case •the ordinary laws of continued locomotion, and in the other •the laws of the propagation of waves among the particles of an elastic fluid.

According to what I have been saying, hypotheses are invented to enable the Deductive Method to be applied to phenomena earlier. But, as I said here, there are three parts to the process of discovering the cause of a phenomenon by the Deductive Method:

(1) induction, to ascertain the laws of the causes;

(2) ratiocination, to compute from those laws how the causes will operate in the particular combination known to exist in the case in hand;

(3) verification, by comparing this calculated effect with the actual phenomenon.

None of these can be dispensed with (though the role of induction may be played by a previous deduction). All the three are found in the deduction proving that gravity is the central force of the solar system. (1) First, it is proved from the moon’s motions that the earth attracts her with a force varying as the inverse square of the distance. This (though partly dependent on previous deductions) corresponds to the first step, the purely inductive one—the ascertainment of the law of the cause. (2) Secondly, from this law together with previously obtained knowledge of the moon’s average distance from the earth and of the actual amount of her deflection from the tangent, it is ascertained how fast the moon would be caused to fall if she were no further off, and no more acted upon by extraneous forces, than terrestrial bodies are; that is the second step, the ratiocination. (3) Finally, this calculated velocity is compared with the observed velocity with which all heavy bodies fall by mere gravity toward the surface of the earth (sixteen feet in the first second, forty-eight in the second, and so forth. . . .) and the two quantities are found to agree. The order in which I have presented the steps was not the order of their discovery, but it’s their correct logical order. . . .

Now the Hypothetical Method suppresses step (1), the induction to ascertain the law; and contents itself with (2) ratiocination and (3) verification; the law that is reasoned from being assumed rather than proved. [This is the first occurrence of ‘Hypothetical Method’ in this work.]

If this process is to be legitimate, the nature of the case must be such that (3) the verification amounts to, and fulfills the conditions of, (1) a complete induction. We want to be assured that the law L that we have hypothetically assumed is a true one; and L’s leading deductively to true results will give this assurance provided the case is such that a false law can’t lead to a true result; provided that no law except L can lead deductively to the conclusions that L leads to. And that is often how things stand. For example, in the deduction that I have just cited the original major premise of the ratiocination, the law of gravitation, was ascertained in this way by this legitimate use of the Hypothetical Method. Newton began by assuming that

the force that at each instant deflects a planet from its straight-line course and makes it curve around the sun is a force tending directly toward the sun.

He then proved that if this is right the planet will mark out (as we know by Kepler’s first law that it does) equal areas in equal times; and lastly he proved that if the force acted in any other direction the planet would not mark out equal areas in equal times. Because this shows that no other hypothesis would square with the facts, the assumption was proved; the hypothesis became an inductive truth. Not only did Newton use this hypothetical process to ascertain the direction of the deflecting force; he also used it to ascertain the law of variation of the strength of that force. He assumed that the force varied inversely as the square of the distance; showed that the remaining two of Kepler’s laws could be deduced from this assumption; and finally showed that any other law of strength-variation would give results inconsistent with those laws, and therefore inconsistent with the real motions of the planets, which Kepler’s laws were known express correctly.

I have said that in this case the verification fulfills the conditions of an induction; but an induction of what sort? On examination we find that it conforms to the canon of the Method of Difference. It presents the two instances

A B C leading to a b c

B C leading to b c.

A represents central force; A B C represents the planetsplus a central force; B C represents the planets with no central force. The planets with a central force give a, areas proportional to the times; the planets without a central force give b c (a set of motions) without a or with something instead of a. This is the Method of Difference in all its strictness. It’s true that the two instances required by the method are obtained not by experiment but by a previous deduction. But that doesn’t matter. It doesn’t matter what the nature is of the evidence from which we derive the assurance that A B C will produce a b c, and that B C will produce only b c; all that matters is that we have that assurance. In this case a process of reasoning provided Newton with the very instances that he would have sought by experiment if the nature of the case had allowed experiments.

So it’s perfectly possible—and in fact quite common—for something that was an hypothesis at the beginning of the inquiry to become a proved law of nature before its close. But for this to happen we must be able, by deduction or experiment, to obtain both the instances that the Method of Difference requires. We can deduce the known facts from the hypothesis, and that gives us the affirmative instance

A B C leading to a b c.

We also have to obtain the negative instance

B C leading to b c,

as Newton did by showing that no antecedent except the one assumed in the hypothesis would in conjunction with B C produce a.

It seems to me that this assurance can’t be obtained if the hypothesis assumes an unknown cause that is imagined solely to account for a. When we are only trying

(i) to determine the precise law of a cause that we have already ascertained or

(ii) to pick out the actual cause from among several agents of the same kind, where we know that one or other of them is the cause

then we can get the negative instance ·that is needed for the Method of Difference·. An example of (ii) would be an inquiry into which of the bodies of the solar system causes by its attraction some particular irregularity in the orbit or periodic time of some satellite or comet. Newton’s inquiry was an example of (i). If it hadn’t already been known that

(ia) the planets were prevented from moving in straight lines by some force tending toward the interior of their orbit, though the exact direction was doubtful;

or if it hadn’t already been known that

(ib) the force increased in some proportion or other as the distance diminished, and diminished as it increased,

Newton’s argument wouldn’t have proved his conclusion. But these facts were already certain; so the range of admissible suppositions was limited to (ia) the various possible directions of a line and (ib) the various possible numerical relations between distance and attractive force. And it was easy to show that different suppositions drawn from this pool couldn’t lead to identical consequences.

So Newton couldn’t have performed his second great scientific operation—identifying terrestrial gravity with the central force of the solar system—by the same hypothetical method. When the law of the moon’s attraction had been proved from the data of the moon itself, then on finding the same law to square with the phenomena of terrestrial gravity he was justified in adopting it as the law of those phenomena likewise; but it wouldn’t have been permissible for him, without any data relating to the moon, to assume that the moon was attracted toward the earth with a force as the inverse square of the distance, merely because that ratio would enable him to account for terrestrial gravity. . . .

So it seems that a really genuinely scientific hypothesis mustn’t be destined always to remain an hypothesis; it must be capable of being either proved or disproved by comparison with observed facts. This is the case •when the effect is already known to depend on the cause that is supposed, and the hypothesis concerns only the precise mode of dependence. . . .; and •when the hypothesis doesn’t concern causation but only the law of correspondence between facts that accompany each other in their variations though there may be no cause-effect relation between them. Kepler’s various false hypotheses about the law of the refraction of light were like that. It was known that the angle at which light came out of the transparent medium varied with every variation in the angle at which it went in, but it wasn’t known what changes in the one corresponded to the different changes of the other. In this case any law different from the true one had to lead to false results. And lastly I should add to these all the hypothetical modes of merely representing or describing phenomena—such as •the hypothesis of the ancient astronomers that the heavenly bodies moved in circles; •the various hypotheses postulating eccentrics, deferents, and epicycles, that were added to that original hypothesis; the nineteen false hypotheses that Kepler made and abandoned concerning the shape of the planetary orbits; and even the doctrine that he finally settled for, that those orbits are ellipses. This was also merely an hypothesis like the rest until it was verified by facts.

In all these cases verification is proof; if the hypothesis squares with the phenomena there’s no need for any other evidence for it. But for that to be the case when the hypothesis relates to causation, I think the supposed cause has not only to be a real phenomenon, something actually existing in nature, but also to be already known to exercise, or at least to be capable of exercising, an influence of some sort over the effect. If that’s not so, the mere fact that we can deduce the real phenomena from •the hypothesis is not sufficient evidence of •its truth.

What if an hypothesis merely assumes a cause, rather than ascribing an assumed law to a known cause? Am I saying that it isn’t scientifically permissible? No. All I’m saying is that it shouldn’t be accepted as true merely because it explains the phenomena. Without being accepted as true, it may usefully suggest a line of investigation that could lead to a real proof; though it can’t even do that (as Comte rightly says) unless the cause it suggests is in its own nature susceptible of being proved by other evidence. This seems to be the philosophical import of Newton’s maxim (so often cited with approval by later writers) that the cause assigned for any phenomenon must not only be one that would explain the phenomenon (if we accepted it) but must also be a vera causa [see Glossary]. Newton didn’t very explicitly define vera causa; and. . . .it’s easy to show that his conception of it was neither precise nor consistent with itself—his optical theory was a striking instance of the violation of his own rule. The cause assigned in an hypothesis certainly doesn’t have to be a cause already known; otherwise we would lose our best opportunities of becoming acquainted with new causes. But what is true in Newton’s maxim is that the cause, though not known previously, should be capable of being known later on—that its existence should be capable of being detected, and its connection with the effect ascribed to it should be capable of being proved by independent evidence. By suggesting observations and experiments, the hypothesis puts us on the road to that independent evidence, if it really is attainable; and until it is actually attained, the hypothesis should be regarded merely as a more or less plausible conjecture.

§5. This function of hypotheses, however, is absolutely indispensable in science. When Newton said Hypotheses non fingo he didn’t mean that he deprived himself of the aid to investigation provided by assuming at first what he hoped eventually to be able to prove. [That is Latin for ‘I don’t make (or invent, make up, contrive, fake) hypotheses’. Fingo is the Latin source for the English word ‘fiction’.] Science could never have reached its present state without such assumptions. They are necessary steps in the progress to something more certain; and nearly everything that is now •theory was once •hypothesis. Even in purely experimental science, there must be some inducement to try one experiment rather than another. It is abstractly possible that all the experiments that have been tried were motivated by the mere desire to discover what would happen in certain circumstances, with no previous conjecture as to the result; but •the experiments that have thrown most light on the general constitution of nature have been unobvious, delicate, and often cumbrous and tedious; •they wouldn’t have had much chance of being undertaken unless there were people who thought that •they could decide whether some general doctrine or theory that had been suggested but not yet proved should be accepted as true. If this is true even of merely experimental inquiry, the conversion of experimental truths into deductive ones was even further from being feasible without large temporary assistance from hypotheses. The process of picking out a regularity in any complicated and seemingly confused set of appearances is bound to be tentative; we begin by making some supposition, even a false one, to see what consequences follow from it; and by seeing how these differ from the real phenomena we learn how to correct our assumption. The simplest supposition that accords with the more obvious facts is the best to begin with, because its consequences are the most easily traced. This rough hypothesis is then roughly corrected, and the operation repeated; and the comparison between •the consequences deducible from the corrected hypothesis and •the observed facts suggests still further corrections, until the deductive results are finally made to tally with the phenomena. . . . As Comte rightly said, neither induction nor deduction would enable us to understand even the simplest phenomena ‘if we didn’t often start by anticipating the results—by making a provisional supposition, at first essentially conjectural, involving some of the very notions that constitute the final object of the inquiry’. Watch how you unravel a complicated mass of evidence; consider, for instance, how you elicit the truth about some event from the involved statements of many witnesses. You’ll find that you don’t take all the items of evidence into your mind at once and try to weave them together; rather, you quickly take a few of the particulars as your basis for a first rough theory about what happened, and then look at the other statements one by one, checking for whether they can be reconciled with your provisional theory, or what alterations or additions it requires to make it square with them. By this procedure, which has been rightly compared to the Methods of Approximation of mathematicians, we arrive through hypotheses at conclusions that aren’t hypothetical.

·START OF A LONG FOOTNOTE·

. . . .The now universally accepted doctrine that the earth is a natural magnet was originally an hypothesis of the celebrated Gilbert.

Another hypothesis (suggested by several recent writers) that isn’t open to any objections and seems likely to light the path of scientific inquiry is that the brain is a kind of electric battery, and that each of its pulsations is a discharge of electricity through the system. It has been noted that the sensation felt by the hand from the pulsing of a brain is very like an electric shock. If this hypothesis is followed to its consequences, it might yield a plausible explanation of many physiological facts, and there’s nothing to discourage the hope that some day we’ll understand electricity well enough to make the truth of the hypothesis checkable against observation and experiment.

When Joseph Gall tried to localise in different regions of the brain the physical organs of our different mental faculties and propensities, this was a legitimate scientific hypothesis; so we ought not to blame him for the extremely slight grounds on which he often proceeded in a scientific project that could only be tentative. We may, however, regret that materials barely sufficient for a first rough hypothesis were hastily worked up into the vain semblance of a science [see ‘Phrenology’ in Wikipedia]. If there really is a connection between the scale of mental endowments and the various degrees of complication in the cerebral system, by far the most likely way to discover that connection is to start with an hypothesis like Gall’s. But because of the special nature of the phenomena, the verification of any such hypothesis faces difficulties that the phrenologists haven’t shown themselves competent even to appreciate, much less to overcome.

Darwin’s remarkable speculation on the Origin of Species is another unimpeachable example of a legitimate hypothesis. What he terms ‘natural selection’ is not only a vera causa but one that has been proved to be capable of producing effects like those that the hypothesis ascribes to it. . . . It is unreasonable to accuse Darwin (as some have) of violating the rules of induction. The rules of induction are concerned with the conditions of proof. Darwin has never claimed that his doctrine was proved. He was bound not by the rules of Induction but by the rules of Hypothesis. And the latter rules have seldom been more completely fulfilled. He has opened a path of inquiry full of promise, the results of which no-one can foresee. And isn’t it a wonderful feat of scientific knowledge and ingenuity to have made such a bold suggestion admissible and discussible when everyone’s first impulse had been to reject it at once, even as a conjecture?

·END OF FOOTNOTE·

§6. It is perfectly consistent with the spirit of the Hypothetical Method to assume in this provisional manner not only an hypothesis •concerning the law of something that we already know to be the cause but an hypothesis •about what the cause is. It is permissible, useful, and often even necessary to begin by asking ourselves what cause may have produced the effect, so that we can know which way to look for evidence to determine whether it actually did. The Descartes’s vortices [see Glossary] would have been a perfectly legitimate hypothesis if there had been the faintest chance that we could ever have a mode of exploration that would enable us to bring it conclusively to the test of observation. The defect of the hypothesis was that it couldn’t lead to any course of investigation that might convert it from an hypothesis into a proved fact. It might be disproved, either by some lack of correspondence with the phenomena it purported to explain or (as actually happened) by some extraneous fact. As Whewell wrote: ‘The free passage of comets through the spaces that these vortices were supposed to inhabit convinced men that the vortices didn’t exist.’ But the hypothesis would have been false even if no such direct evidence of its falsity had been available. Direct evidence of its truth there could not be.

The prevailing hypothesis of a luminiferous ether [see Glossary] is not entirely cut off from the possibility of direct evidence in its favour (that being the main difference between it and Descartes’s hypothesis about vortices). It’s well known that the difference between •the calculated times of the periodical return of Encke’s comet and •and the observed times has led to a conjecture that something that can resist motion is diffused through space. If this surmise were confirmed by the gradual accumulation, through the centuries, of a similar variance in the case of the other bodies in the solar system, the luminiferous ether would have gone a long way toward being a vera causa, because we’d have established that there is a great cosmical agent that has some of the attributes assumed by the hypothesis; though many ·of the •old· difficulties would remain, and I imagine that there would also be •new ones arising from the identification of the ·previously hypothesised· ether with the ·more recently discovered· resisting medium. At present, however, this hypothesis can’t be regarded as more than a conjecture; the existence of the ether still rests on the possibility of deducing from its assumed laws a considerable number of actual phenomena; and I can’t regard this evidence as conclusive, because we can’t be sure that if the hypothesis is false it must lead to results at variance with the true facts.

Accordingly, most sober thinkers accept that an hypothesis of this kind isn’t to be accepted as probably true because it accounts for all the known phenomena. Sometimes two conflicting hypotheses account for all the known phenomena; and there are probably many others that are equally possible though they don’t come into our minds because there’s nothing analogous to them in our experience. But here’s something that many people seem to think:

An hypothesis of the kind we’re considering is entitled to a more favourable reception if, besides •accounting for all the facts previously known, it •leads to the anticipation and prediction of others that experience later verifies—as the wave theory of light led to the prediction, subsequently confirmed by experiment, that two luminous rays might meet each other in such a way as to produce darkness.

You might expect that from a layman; but people with scientific attainments also—strangely!—lay stress on the fulfillment of this kind of prediction. If the laws of the propagation of light square with the laws of vibration in an elastic fluid in as many respects as is needed to make the hypothesis provide a correct expression of most of the phenomena known at the time, it’s not surprising if they agree in one respect more. Even twenty such agreements wouldn’t prove the reality of an ether in which waves occur; it wouldn’t follow •that the phenomena of light were results of the laws of elastic fluids, but at most •that they’re governed by laws that overlap with these. . . .[2] Even in our imperfect acquaintance with nature we can cite cases where agencies that we have good reason to consider as radically distinct produce some or all of their effects according to laws that are identical. The law of the inverse square of the distance is the measure of the intensity not only of •gravitation but also (we think) of •illumination and of •heat diffused from a centre. Yet no-one thinks that because these three kinds of phenomena •obey the same law they are therefore •produced by the same mechanism.

[Mill quotes Whewell •disagreeing with the line Mill has been taking here and •illustrating his position with a peculiar example; Mill’s response to this is also peculiar, and we can afford to by-pass this exchange. After it, Mill gets back on track:] The agreement of •the phenomena of light with •the theory of light-waves must arise from ·overlap, i.e. from· the actual identity of some of the laws of waves with some of those of light. . . . But from the fact that some of the laws ·of light· agree with the laws of waves it doesn’t follow that there are any actual light-waves; any more than it followed from the fact that some (though not so many) of the laws of light agreed with the laws of the projection of particles that there was actual emission of particles. Even the light-waves hypothesis doesn’t account for all the phenomena of light. •The natural colours of objects, •the compound nature of the solar ray, •the absorption of light, and •its roles in chemical and vital action—the hypothesis leaves these as mysterious as it found them. And some of these facts seem to fit better with particle theory than with the ·wave· theory of Young and Fresnel. For all we know, some third hypothesis will in time leave the wave theory as far behind as it has left the ·particle· theory of Newton and his successors.

I have said that ‘Hypothesis H accounts for all the known phenomena’ is often equally true of two conflicting hypotheses; and Whewell has remarked that he knows ‘of no such case in the history of science, where the phenomena are at all numerous and complicated’. . . . But a few pages earlier he carefully refuted this by maintaining that all or most exploded scientific hypotheses could have been modified so as to make them correct representations of the phenomena. The hypothesis of vortices, he tells us, went through a series of modifications until its results coincided with •those of the Newtonian theory and with •the facts. Actually, the vortices didn’t explain all the phenomena that the Newtonian theory was eventually found to account for—e.g. they didn’t explain •the precession of the equinoxes—but •this phenomenon was not something that either side in the dispute had in mind as needing to be accounted for. We can believe on Whewell’s authority that all the facts those people did have in mind accorded as accurately with the Cartesian hypothesis, in its finally improved state, as with Newton’s.

But even if hypothesis H accounts for the facts and we can’t imagine any other that does so, that isn’t a valid reason for accepting H. There’s no need to suppose that the true explanation must be one that we—with our limited experience so far—could imagine. Among the natural agents we’re acquainted with, the vibrations of an elastic fluid may be the only one whose laws are like the laws of light; but for all we know there may be an unknown cause that •is not an elastic ether diffused through space yet •produces effects identical in some respects with the effects that waves in such an ether would produce. To assume that no such cause can exist ·because we can’t at present conceive it· appears to me an extreme case of assumption without evidence. And. . . . I can’t help expressing astonishment that a philosopher of Whewell’s abilities and attainments should have written an elaborate treatise on the philosophy of induction in which he recognises absolutely no mode of induction except that of

trying hypothesis after hypothesis until one is found that fits the phenomena; which one, when found, is to be assumed as true, with only one reservation, namely if on re-examination it turns out to assume more than is needed for explaining the phenomena, the superfluous part of it should be cut off.

And this without any distinction between cases where •it can be known in advance that two hypotheses can’t lead to the same result, and cases where •for all we know to the contrary there may be an infinity of hypotheses that are consistent with the phenomena.[3]

But I don’t join Comte in condemning those who labour to work out in detail the application of these hypotheses to the explanation of ascertained facts, provided they bear in mind that the most they can prove is not that the hypothesis is true but that it may be true. The ether hypothesis has a very strong claim to be followed out in that way, a claim that was greatly strengthened when it was shown to provide a mechanism that would explain the mode of production of heat as well as of light. Indeed, the theory has a smaller element of hypothesis in its application to heat than in the case for which it was originally formed. We have proof by our senses that there is. . . .movement among the particles of all heated bodies, while we have nothing analogous to that in the case of light. Thus, when heat is communicated from the sun to the earth across apparently empty space, the chain of causation has. . . .motion at the beginning and at the end. The hypothesis only makes the motion continuous by extending it to the middle. Now, we know that motion in a body can be passed on to another body contiguous to it; and the intervention of a hypothetical elastic fluid occupying the space between the sun and the earth provides the contiguity that is the only thing missing—and can’t be supplied without an intervening medium. Still, the supposition is at best a probable conjecture, not a proved truth. For there’s no proof that contiguity is absolutely required for motion to be passed from one body to another. Contiguity doesn’t always exist, to our senses at least, in cases where motion produces motion. The forces that go under the name of ‘attraction’, especially the greatest of all, gravitation, are examples of motion producing motion apparently without contiguity. When a planet moves, its distant satellites move with it. The sun carries the whole solar system with it in the progress it is making through space. Some theorists have come up with the geometrical reasonings (like the ones the Cartesians used to defend their vortices) by which they have tried to show that the motions of the ether can account for gravitation itself; but even if we accepted this as conclusive, it wouldn’t follow that this is the mechanism of gravitation.

§7. Before leaving the topic of hypotheses, I should guard against the appearance of questioning the scientific value of several branches of physical inquiry which, though only in their infancy, I regard as strictly inductive. There’s a great difference between (i) inventing agencies to account for classes of phenomena, and (ii) trying in conformity with known laws to conjecture what earlier collocations [see Glossary] of known agents may have given rise to individual facts that are still in existence. Of these, (ii) is the legitimate operation of inferring from an observed effect E the past existence of a cause similar to the cause that we know produced E in all the cases where we have actual experience of its origin. That’s what goes on in the inquiries of geology; and they are no more illogical or fanciful than judicial inquiries that also aim at discovering a past event by inference from its present effects. Just as

we can ascertain whether a man was murdered or died a natural death, from the state of the corpse, the presence or absence of signs of struggle, the marks of blood, the footprints of the supposed murderers and so on, relying all the way on uniformities ascertained by a perfect induction with no hypothesis stirred into the mix,

so also

if we find on and beneath the surface of our planet masses exactly like deposits from water, or like results of the cooling of matter that has been melted by fire, we’re entitled to conclude that that was their origin; and if the effects, though similar in kind, are on a far larger scale than any which are now produced, we may rationally—and without hypothesis—conclude that the causes existed formerly with greater intensity or operated during an enormous length of time.

Since the rise of the present enlightened school of geological theorising, no geologist of authority has tried to go further than this.

In many geological inquiries it doubtless happens that though the laws to which the phenomena are ascribed are known laws, and the agents are known agents, those agents are not known to have been present in the particular case. In the theory that granite began as molten lava, there’s no direct proof that this substance ever was actually subjected to intense heat. But the same thing could be said of all judicial inquiries that go by circumstantial evidence. We can conclude that a man was murdered, without its being proved by the testimony of eye-witnesses that someone who had intended to murder him was present on the spot. For most purposes it’s enough if no other known cause could have generated the effects that have been found.

Laplace’s famous theory about the origin of the earth and planets is essentially based on inductive procedures like those of modern geological theory. The theory is this:

The sun’s atmosphere originally extended to the present limits of the solar system; by cooling, it contracted to its present size; that shrinkage of the sun and its atmosphere made them spin ever faster (this is guaranteed by the general principles of mechanics); the increased centrifugal force generated by the faster rotation counteracted gravity and caused the sun to abandon successive rings of vapourous matter; these condensed by cooling and became the planets.

This theory doesn’t hypothetically •introduce any unknown substance or •attribute any unknown property or law to a known substance. The known laws of matter authorise us to suppose that a body that constantly gives out as much heat as the sun does must grow steadily cooler, which must make it contract; so the present state of affairs requires us to suppose that the sun’s atmosphere used to extend much further than it does now; and we’re entitled to suppose that it extended as far out as we can find effects of the sort it might naturally leave behind it on shrinking; and that’s what the planets are. [Mill continues to spell out all the steps needed by Laplace’s theory, and concludes:] So Laplace’s theory contains nothing that is strictly speaking hypothetical; it’s an example of legitimate reasoning from a present effect to a possible past cause, according to the known laws of that cause. Although I have likened this to the theories of geologists, it is considerably less secure than them.. . . . There is a much greater chance of error in assuming that the present laws of nature are the same ones that operated at the origin of the solar system than in merely presuming (with geologists) that those laws have lasted through a few revolutions and transformations of a single one among the bodies of which that system is composed.

NOTES

[1] [In this note Mill approvingly quotes a long passage from Bain, •saying that similarities between phenomena offer hope of uniting their laws; •pointing out that gravitational attraction is strikingly similar to the cohesion or holding-together of bodies; and then •insisting that there is nevertheless no chance of theoretically uniting those two kinds of force. The quotation continues:] ‘The two kinds of force agree in the one point, attraction, but they agree in no other; indeed in the manner of the attraction they differ widely. . . . Gravity is common to all matter, and equal in amount in equal masses of matter, whatever be the kind; it follows the law of the diffusion of space from a point (the inverse square of the distance); it extends to distances unlimited; it is indestructible and invariable. Cohesion is special for each separate substance; it decreases according to distance much more rapidly than the inverse square, vanishing entirely at very small distances. Two such forces aren’t alike enough to be generalised into one force; the generalisation is only illusory; the statement of the difference would still make two forces, while the consideration of one wouldn’t in any way simplify the phenomena of the other, as happened in the generalisation of gravity itself.’ To the impassable limit of the explanation of laws of nature that I expounded in the text we must therefore add a further limitation. When the phenomena to be explained are not in their own nature generically distinct ·like colour and odour·, the attempt to refer them to the same cause is scientifically legitimate; but for the attempt to succeed, the cause must be shown to be capable of producing the phenomena according to the same law. Otherwise the unity of cause is a mere guess, and the generalisation only a nominal one which, even if accepted, wouldn’t lower the number of ultimate laws of nature.

[2] What has contributed most to the acceptance of the hypothesis of a physical medium for the conveyance of light is a trio of facts: •that light travels (which can’t be proved of gravitation), •that its communication is not instantaneous, but takes time; and •that it is intercepted by intervening objects (which gravitation is not). These are respects in which the phenomena of light fit those of the mechanical motion of a solid or fluid substance. But we aren’t entitled to assume that mechanical motion is the only power in nature that can exhibit those attributes.

[3] Whewell has recently made a concession regarding the medium of the transmission of light that removes the difference between us, but I can’t make sense of it in the context of the rest of his doctrine on this subject. Arguing that all matter has weight, he cites Hamilton’s reference to the luminiferous ether and the calorific and electric fluids ‘which we can neither •denude of their status as substance nor •clothe with the attribute of weight’. Whewell comments: ‘My reply is that precisely because I can’t clothe these agents with the attribute of weight I do denude them of the status of substance. They aren’t substances, they are agencies. These weightless agents aren’t properly called weightless fluids! I think I have proved this.’ Nothing can be more philosophical. But if the luminiferous ether isn’t matter—indeed if it isn’t fluid matter—what is the meaning of the waves in it? Can an agency undulate? Can there be alternate motion forward and backward of the particles of an agency? And doesn’t the whole mathematical theory of waves imply that they are material?. . . .

Chapter 15. Progressive effects. The continued action of causes

§1. In chapters 11–14 I have traced the general outlines of the theory of the generation of derivative laws from ultimate ones. In this chapter I’ll deal with one particular kind of derivation of laws from other laws—a kind that is so general and so important that it demands a separate examination.

Some phenomena, e.g. some bodily sensations, are essentially instantaneous; their existence can be prolonged only by prolonging the existence of the cause that produces them. Most phenomena, however, are in their own nature permanent: having begun to exist, they would exist forever unless some cause intervened with a tendency to alter or destroy them. Such, for example, are all the facts of phenomena that we call ‘bodies’. Once water has been produced, it won’t of itself relapse into a state of hydrogen and oxygen; such a change requires some agent that can decompose the compound. Similarly with bodies’ positions and movements in space, No object at rest starts moving without the intervention of something external to itself; and no object, once it is moving, returns to a state of rest or alters its direction or velocity unless some new external factor comes into play. So it perpetually happens that a temporary cause gives rise to a permanent effect. (a) A few hours of contact between iron and moist air produces a rust that may last for centuries; (b) a force that launches a cannon-ball into space produces a motion that would continue forever if no other force counteracted it.

Between those two examples there’s a difference worth pointing out. In (a) (in which the effect is a substance and not a motion of a substance), since the rust remains unaltered unless some new cause intervenes, we can speak of the contact of moist air a century ago as the immediate cause of the rust that has existed from then until now. But when the effect is motion, which is itself a change, we must use a different language. The permanence of the effect is now only the permanence of a series of changes. The second inch or foot or mile of motion is not the mere prolonged duration of the first inch or foot or mile; it is another fact that follows the other and may in some respects be very unlike it because it carries the body through a different region of space. The original force that set the body moving is the remote cause of all its motion, however long that is continued; but it is the immediate cause only of the motion that occurred at the first instant. The motion at any subsequent instant is immediately caused by the motion that occurred at the preceding instant, and not on the original moving cause. . . . This is recognised by mathematicians when they include the force generated by the motion of a body at t_1 among the causes of its motion at t_2. This would be absurd if it meant that this ‘force’ was an intermediate link between the cause and the effect. What it in fact refers to is only the previous motion itself, considered as a cause of further motion. So if we want to speak with perfect precision, we should consider each •link in the succession of motions as the effect of the •link preceding it. But if we find it convenient to speak of the whole series as one effect, it must be as an effect produced by the original push: a permanent effect produced by an instantaneous cause, and having the property of self-perpetuation.

Now consider the situation when the original agent or cause is itself permanent. Whatever effect has been produced up to a given time would (unless prevented by the intervention of some new cause) exist permanently, even if the cause were to perish. But the cause doesn’t perish but continues to exist and to operate; so it must go on producing more and more of the effect; and instead of •one uniform effect we have •a growing series of effects arising from the accumulated influence of the permanent cause. Iron’s contact with the atmosphere causes part of it to rust; and if the iron were then protected from the atmosphere that rust would be permanent but no more would be added. If the iron continues to be exposed to moist air, rusting continues until all the exposed iron is converted into a red powder. . . . Another example: the existence of the earth at a given instant causes an unsupported body to move towards it at the next instant; and if the earth were annihilated the effect already produced would continue—the object would move in the same direction with its acquired velocity until intercepted by some body or deflected by some force. But the earth isn’t annihilated, so it goes on producing in each ‘next instant’ an effect similar (in kind and quantity) to the effect in the preceding instant. The addition of these two effects to one another results in an accelerated velocity; and as this operation is repeated at each successive instant, the mere permanence of the cause—without any increase of it—gives rise to a continual increase of the effect, so long as all the conditions, negative and positive, of the production of that effect continue to exist.

Obviously this state of affairs is merely a case of the composition of causes. A cause that continues in action must on a strict analysis be considered as

a number of exactly similar causes, successively introduced and jointly producing the sum of the effects that they would separately produce if they acted singly.

Strictly speaking, the progressive rusting of the iron is the sum of the effects of many particles of air acting in succession on corresponding particles of iron. The earth’s continued action on a falling body is equivalent to a series of forces applied in successive instants, each tending to produce a certain constant quantity of motion; and the motion at t2 is the sum of •the effects of the new force applied at t1 and •the motion that had already been acquired before that. . . . The effect produced by the earth’s influence at the most recent instant is added to the sum of the effects whose remote causes were the influences exerted by the earth at all the previous instants since the motion began. So this case comes under the principle of a concurrence of causes producing an effect equal to the sum of their separate effects.But because

•the causes come into play successively, and

•the effect at each instant is the sum of the effects of only the causes that have come into action up to that instant,

the result takes the form of an ascending series—a series of sums, each greater than its immediate predecessor—and this gives us a progressive effect from the continued action of a cause.

The continuance of the cause influences the effect only by adding to its quantity, and the addition conforms to a fixed law (equal quantities in equal times); so the result can be calculated mathematically. In fact, this case of infinitesimal increments is precisely what the differential calculus was invented to meet. The questions

•what effect will result from the continual addition of a given cause to itself?

•what amount of the cause, being continually added to itself, will produce a given amount of the effect?

are obviously mathematical questions, and therefore to be treated deductively. We have seen that compositions of causes are seldom fit for anything but deductive investigation, and this is especially true in our present case of the continual composition of a cause with its own previous effects. Why? Well, this is especially amenable to the deductive method, and is bound to elude experimental treatment because of how gradually the effects are blended with one another and with the causes.

§2. I come now to a more intricate case of the composition of causes, namely the case where the cause doesn’t merely •continue in action but •undergoes a continuous change in respects that are relevant to the effect. Here as before, the total effect goes on accumulating by the continual addition of fresh effects to those already produced, but now it’s not by adding equal quantities in equal times; the •quantities added are unequal, and even the •quality may now be different. If the change in the state of the permanent cause is progressive, the effect will go through a double series of changes—•partly from the accumulated action of the cause, and •from the changes in its own action. The effect is still a progressive effect, but this time produced not by the mere continuance of a cause but by its continuance and its progressiveness combined.

A familiar example: the increase of the temperature as summer advances, i.e. as the sun draws nearer to a vertical position and remains for more hours above the horizon. . . . When the sun has come near enough to the zenith, and remains above the horizon long enough, to give more warmth during one daily rotation than the earth’s radiation can remove, the mere continuance of the cause would progressively increase the effect, even if the sun came no nearer and the days grew no longer; but in addition to this a change takes place because of the increase in the amount of heat the sun sends to us because of its changing position in the sky. When the summer solstice has passed, the progressive change in the cause begins to go in the opposite direction, but for a while the accumulating effect of the mere continuance of the cause exceeds the effect of the changes in it, and the temperature continues to rise.

A planet’s motion is a progressive effect, produced by causes that are both permanent and progressive. The planet’s orbit is determined. . . .by two causes:

(i) the action of the sun, a permanent cause that •alternately increases or diminishes as the planet comes to be nearer to or further from the sun, and •acts in a different direction at every point; and

(ii) the planet’s tendency to continue moving in the direction and with the velocity that it has already acquired. This force also grows greater as the planet draws nearer to the sun because it speeds up as it does so and slows down as it recedes from the sun; and it also acts in a different direction at each point, because at every point the sun’s action in deflecting the planet from its previous direction alters the line in which it tends to continue moving.

The planet’s motion at t_2 is determined by •the amount and direction of its motion, and •the amount and direction of the sun’s action on it, at t_1; and if we speak of the planet’s entire journey around the sun as one phenomenon (which. . . .we often find it convenient to do), that phenomenon is the progressive effect of two permanent and progressive causes, the sun’s force and the acquired motion. Those causes happen to be progressive in the special way that we call ‘periodical’, so the effect has to be periodical too. . . .

Another feature of this example is worth thinking about. Though the causes themselves are permanent and independent of all conditions known to us, the changes in the quantities and relations of the causes are actually caused by changes in the effects. [Mill explains this in more detail than we need. The point is just that the difference between

•the strength and direction of the sun’s pull on the planet at time t_1 and •the strength and direction of its pull at t_2

and also the difference between

•the strength and direction of the planet’s tendency to move in a straight line at t_1 and •the strength and direction of its tendency to move in a straight line at t_2

are both caused by facts about how the planet moves between those two times.]

§3. In all cases of progressive effects, whether arising from the accumulation of unchanging or of changing elements, there is a uniformity of succession not merely between the cause and the effect, but between the first stages of the effect and its subsequent stages. . . . The sequence of spring and summer is regular and invariable. . . ., but we don’t consider spring to be the cause of summer; it’s evident that both are successive effects of the heat received from the sun, and that spring considered merely in itself could continue for ever without having the slightest tendency to produce summer. As I have so often remarked, the cause is the unconditional invariable antecedent. . . .

This is how most uniformities of succession are generated—I mean ones that aren’t cases of causation. When a phenomenon goes on increasing, or periodically increases and diminishes, or goes through any continued and unceasing process of variation reducible to a uniform rule or law of succession, we don’t infer from this that any two neighbours in the series are cause and effect. We presume the contrary; we expect to find that the whole series originates either from •the continued action of fixed causes or from •causes that are themselves continuously changing. A tree grows from half an inch high to a hundred feet; and trees of some species will generally grow to that height unless prevented by some counteracting cause. But we don’t call the seedling the cause of the full-grown tree; it certainly is the invariable antecedent, and we don’t know much about what other antecedents the sequence depends on, but we’re convinced that it depends on something. Why? Because. . . .the close resemblance of the seedling to the tree in all respects but size, and the gradualness of the growth, so exactly resemble the progressively accumulating effect produced by the long action of some one cause that we can’t possibly doubt that the seedling and the tree are two terms in a series of that sort, the first term of which we haven’t yet found. The conclusion is further confirmed by the fact that we can prove by strict induction that the tree’s growth, and even its continued existence, depend on the continued repetition of certain processes of nutrition—the rise of the sap, the absorptions and exhalations by the leaves, etc.—and the same experiments would probably prove to us that the growth of the tree is the accumulated sum of the effects of these continued processes if it weren’t for the fact that our eyesight isn’t microscopic enough for us to observe correctly and in detail what those effects are.

In such a case the effect may during its progress undergo many modifications besides those of quantity, and may sometimes appear to undergo a very marked change of character. This could be because •the unknown cause consists of several component elements whose effects, accumulating according to different laws, are compounded in different proportions at different times; or because •at certain points in the effect’s progress fresh causes or agencies come in, or are evolved, which intermix their laws with those of the primary agent.

Chapter 16. Empirical laws

§1. When observation or experiment has shown that a uniformity U exists, but scientists can’t see any reason why U exists and therefore hesitate to rely on it in cases varying much from those that have been actually observed, they call U an Empirical Law. In calling something an empirical law we imply that it’s not an ultimate law—that if it is true, its truth can be and should be accounted for. It is a derivative law, the derivation of which is not yet known. To state the explanation, the why, of the empirical law would be to state the laws from which it is derived—the ultimate causes on which it depends; and if we knew these we would also know what its limits are, i.e. under what conditions it would cease to be fulfilled.

The periodic return of eclipses, as originally ascertained by the early Eastern astronomers’ many observations, was an empirical law until the general laws of the celestial motions had accounted for it. The following are empirical laws still waiting to be resolved into the simpler laws from which they are derived:

•the local laws of the rise and fall of the tides in different places;

•the relation between certain kinds of weather and certain appearances of sky;

•the apparent exceptions to the almost universal truth that bodies expand by increase of temperature;

•the law that animal and vegetable species are improved by cross-breeding:

and also the fact that

•gases have a strong tendency to permeate animal membranes;

•substances with a very high proportion of nitrogen (such as hydrocyanic acid and morphia) are powerful poisons;

•when different metals are fused together the alloy is harder than the various elements;

•the number of atoms of acid required to neutralise one atom of any base is equal to the number of atoms of oxygen in the base;

•the solubility of substances in one another depends, at least in some degree, on the similarity of their elements.[1]

An empirical law, then, is an observed uniformity, presumed to be resolvable into simpler laws but not yet resolved into them. Empirical laws are often discovered long before they are explained by the Deductive Method; and the verification of a deduction usually consists in comparing its results with empirical laws previously ascertained.

§2. A small number of ultimate laws of causation generates a vast number of derivative uniformities, both of succession and of coexistence. (a) Some are laws of succession or coexistence between different effects of the same cause; I gave examples of these it chapter 15. (b) Some are laws of succession between effects and their remote causes, resolvable into the laws connecting each with the intermediate link. (c) When causes act together and compound their effects, the laws of those causes generate the fundamental law of the effect, namely that it depends on the coexistence of those causes. (d) Finally, the order of succession or of coexistence that holds among effects necessarily depends on their causes. If they are effects of a single cause, it depends on the laws of that cause; if they’re effects of different causes, it depends on the laws of those causes separately and on the circumstances that determine their coexistence. If we investigate when and how the causes will coexist, that depends on their causes; and we may thus trace back the phenomena higher and higher until

•the different series of effects meet in a point, and the whole thing is shown to have depended ultimately on some common cause;

or until

•instead of converging to one point they terminate in different points, and the order of the effects is proved to have arisen from the collocation of some of the ultimate causes.

For example, the order of succession and of coexistence among the heavenly motions that Kepler’s laws express is derived from the coexistence of two primeval causes, •the sun and •each planet’s original impulse or projectile force. Kepler’s laws are resolved into the laws of these causes and the fact of their coexistence.

So derivative laws don’t depend solely on the ultimate laws into which they are resolvable; they mostly depend on •those ultimate laws and •an ultimate fact, namely the mode of coexistence of some of the elements of the universe. The ultimate laws of causation could be just what they actually are and yet the derivative laws completely different, if the causes coexisted in different proportions or with any difference in such of their relations as influence the effects. If, for example, •the sun’s attraction and •the original projectile force had existed in some other ratio to one another than they did (and we know of no reason why this couldn’t have been the case), the derivative laws of planetary motions could have been quite different from what they are. The ratio that does exist happens to be such as to produce regular elliptical motions; any other ratio ·of sun’s attraction to original projectile force· would have produced different ellipses, or circles, parabolas, or hyperbolas, but still regular trajectories because the effects of each of the agents accumulate according to a uniform law; and two regular series of quantities, when their corresponding terms are added, must produce a regular series of some sort. . . .

§3. Now this last-mentioned element in the resolution of a derivative law—the element that is not a law of causation but a collocation of causes—can’t itself be reduced to any law. As I remarked here, no uniformity or norm or principle or rule is perceivable in the distribution of the primeval natural agents through the universe. The different substances composing the earth stand in no constant relation to the powers that pervade the universe. One substance is more abundant than others, one power acts through a larger extent of space than others, without any pervading analogy that we can discover. We don’t know why •the sun’s attraction and •the force in the direction of the tangent coexist in the exact proportion they do, and we can’t trace any match between the sun’s attraction and the proportions in which any other elementary powers in the universe are intermingled. The utmost disorder in the combination of the causes is consistent with the most regular order in their effects; because when each agent acts according to a uniform law even the most capricious combination of agencies will generate a regularity of some sort; as we see in the kaleidoscope, where any casual arrangement of coloured bits of glass produces through the laws of reflection a beautiful regularity in the effect.

§4. This justifies the attitude of scientists in not relying much on empirical laws.

A derivative law that results wholly from the operation of some one cause will be as universally true as the laws of the cause are—i.e. it will always be true except where some one of its effects is defeated by a counteracting cause. But when the derivative law results not from •different effects of one cause but from •effects of several causes, we can’t be certain that it will remain true if there’s some re-arrangement of those causes or of the primitive natural agents on which the causes ultimately depend. The proposition that coal-beds always rest on strata of kind K, though true on the earth as far as we know, can’t be extended to the moon or the other planets, supposing that they have coal, because we can’t be sure that the initial constitution of any other planet was such as to lay down geological deposits in the same order as on our globe. The derivative law in this case depends not only on laws but also on a collocation [see Glossary]; and collocations can’t be reduced to any law.

If EL is an empirical law—i.e. a derivative law that hasn’t yet been resolved into its elements—then of course we don’t know whether it results from •the different effects of one cause, or from •effects of different causes. We can’t tell whether it depends wholly on laws, or partly on laws and partly on a collocation. If EL depends on a collocation, it will be true in every case where that particular collocation exists. But we don’t know, supposing it does depend on a collocation, what the collocation is; so we aren’t safe in extending EL beyond the limits of time and place where we have actual experience of its truth. Since it has always been found true within those limits, we have evidence that the relevant collocations, whatever they are, really do exist within those limits. But we have no basis for inferring that because a collocation is proved to exist within certain limits of place or time it will exist beyond those limits. So empirical laws can be accepted as true only within the limits of time and place in which they have been found true by observation; indeed, only within the limits of time and place and circumstance; for we don’t know the ultimate laws of causation on which EL depends, so we can’t foresee, without actual trial, how the introduction of any new circumstance may affect it.

§5. How are we to know that a uniformity ascertained by experience is only an empirical law? We haven’t been able to resolve it into any other laws, so how do we know that it isn’t itself an ultimate law of causation?

I answer that no generalisation amounts to more than an empirical law if the only support for it comes from the Method of Agreement. We have seen that we can never arrive at causes by that method alone. The utmost that the Method of Agreement can do is to ascertain all the circumstances common to all cases in which a phenomenon P is produced; and this aggregate includes not only •the cause of P but all the phenomena P is connected with by any derivative uniformity, whether as collateral effects of a single cause or as effects of some other cause that has coexisted with it in all the instances we have observed. The Method of Agreement doesn’t offer any way of determining which of these uniformities are laws of causation and which are merely derivative laws resulting from the laws of causation and the collocation of the causes. So none of them can be accepted as anything but derivative laws whose derivation hasn’t been traced—i.e. empirical laws. And that’s the status we must assign to all results obtained by the Method of Agreement (and therefore almost all truths obtained by simple observation without experiment), until they are confirmed by the Method of Difference or explained deductively.

These empirical laws may have (i) more or (ii) less authority, depending on whether there’s reason to think that they are resolvable into (i) laws only or into (ii) laws and collocations together. (i) The sequences that we observe in the production and subsequent life of an animal or a vegetable, resting purely on the Method of Agreement, are mere empirical laws; but though the antecedents in those sequences may not be the causes of the consequents, all the stages in the sequences are doubtless successive stages of a progressive effect of a common cause, and are therefore independent of collocations. (ii) On the other hand, the uniformities in the top-to-bottom order of strata on our planet are empirical laws of a much weaker kind, because as well as not being laws of causation there’s no reason to believe that they depend on any common cause; all appearances are in favour of their depending on a particular collocation of natural agents that existed on our globe at some time or other—a collocation that there’s no reason to think supports any inference about what collocation does or did exist in any other part of the universe.

§6. My definition of ‘empirical law’ made that phrase applicable not only to uniformities that aren’t known to be laws of causation, but also to ones that are, provided there’s reason to presume that they aren’t ultimate laws. Now is the time to ask: By what signs can we judge, of an observed uniformity that ·we are satisfied· is a law of causation, that it is a derivative and not an ultimate law? ·I shall present two such signs·.

(1) The first sign is one we get if there’s evidence that between the antecedent a and the consequent b there’s some intermediate link, some phenomenon that we can guess exists there, though our senses or our instruments aren’t sharp enough for us to ascertain its precise nature and laws. If there is such an intermediate phenomenon IP, it follows that even if a is the cause of b, it is only the remote cause, and that the law a causes b is resolvable into at least two laws, a causes IP and IP causes b. This is a very common case, because the operations of nature are mostly on such a minute scale that many of the successive steps can’t be clearly perceived if indeed they are perceived at all.

Consider the laws of the chemical composition of substances—e.g. that when hydrogen and oxygen are combined water is produced. All we see of the process is

•the two gases are mixed in certain proportions,

•heat or electricity is applied,

•an explosion takes place,

•the gases disappear, and

•water remains.

There’s no doubt about the law, or about its being a law of causation. But between the antecedent (the gases in a state of mechanical mixture, heated or electrified) and the consequent (the production of water) there must be an intermediate process that we don’t see. For if we analyse any portion of the water, we find that it always contains hydrogen and oxygen—indeed, in the very same proportions. This is true of a single drop; it’s true of the smallest portion our instruments can evaluate. And since the smallest perceptible portion of the water contains both those substances, portions of hydrogen and oxygen smaller than the smallest perceptible must have come together in every minute portion of space; must have come closer together than when the gases were merely mechanically mixed since (to mention just one reason) the water occupies far less space than the gases. Now, we can’t see this contact or close approach of the tiny particles, so we can’t observe what circumstances accompany it or what laws are at work when it produces its effects. The production of water. . . .may be a very remote effect of those laws. There may be countless intervening links, and we are sure that there must be some. We have full proof that each of the great transformations in the sensible properties of substances is preceded by some kind of corpuscular [see Glossary] action; so we can’t doubt that the laws of chemical action, as at present known, are not ultimate laws but derivative ones—even if we will never know the nature of the laws of corpuscular action they are derived from.

Similarly, all the processes of vegetative life, whether in plants or in animals, are corpuscular processes. Nutrition is the addition of particles to one another, sometimes merely replacing other particles that have been separated and excreted, sometimes adding to the organism’s size or weight, but doing this so gradually that it isn’t perceptible until it has gone on for a long time. Various organs have their own special vessels in which they store fluids whose component particles must have been in the blood, though they are utterly unlike blood in their mechanical properties and in their chemical composition. Here’s an abundance of unknown links to be filled in; and there can’t be any doubt that the laws of the phenomena of living organisms are derivative laws, dependent on properties of •corpuscles and of •elementary tissues that are comparatively simple combinations of corpuscles.

(2) We encounter the second sign that a law isn’t ultimate when its antecedent A is an extremely complex phenomenon, which makes it likely that A’s effects are at least partly compounded out of the effects of A’s different elements. Cases where the effect of the whole is not made up of the effects of its parts are rare ·and therefore unlikely·, so the Composition of Causes is by far the more ordinary ·and thus more probable· case.

I’ll illustrate this by two examples. In the first, the antecedent is the sum of many homogeneous parts. The weight of a body is made up of the weights of its tiny particles; and astronomers avail themselves of this when they say that bodies at equal distances gravitate towards one another in proportion to their quantity of matter—·implying that what holds for the big things also holds for the little ones·. So all true propositions concerning gravity are derivative laws; the ultimate law into which they are all resolvable is that every particle of matter attracts every other. In my second example, the antecedent is the sum of many heterogeneous parts. Let it be any one of the sequences observed in meteorology—e.g. the fact that a lessening of atmospheric pressure (indicated by a fall of the barometer) is followed by rain. The antecedent is here a complex phenomenon, made up of heterogeneous elements: the column of the atmosphere over any particular place consists of two parts—a column of air and a column of water-vapour mixed with it—and the change in these two together that is •shown by a fall of the barometer and •followed by rain must be a change in the air or a change in the water-vapour or a change in both. So even if that’s all we have to go on, we can reasonably suppose—given the invariable presence of both these elements in the antecedent—that the sequence is probably not an ultimate law, but a result of the laws of air and the laws of water-vapour. If we come to know those laws so well that we’re in a position to say that they couldn’t, unaided, produce the observed results in the barometer and the weather, then of course we must give up this supposition. But not until then.

In almost all known cases in which a very complex antecedent A is regularly followed by a state of affairs S, we can either

•actually account for the sequence A → S in terms of simpler laws, or

•infer with great probability (from our knowledge that there are intermediate causal links though we don’t know what they are) that A → S could be accounted for in that way.

So it’s highly probable that all sequences from complex antecedents are resolvable like that, and that ultimate laws are all comparatively simple. If we didn’t have the reasons that I gave here for believing that the laws of organised nature are resolvable into simpler laws, it would be almost a sufficient reason that the antecedents in most of the sequences are so very complex.

§7. I have recognised two kinds of empirical laws—•those known to be laws of causation, but presumed to be resolvable into simpler laws, and •those not known to be laws of causation at all. These two have several things in common:

•They both agree in the demand they make for being explained by deduction.

•They are both appropriate means of verifying such a deduction, because they represent the experience that the result of the deduction must be compared with.

•Until they are explained and connected with the ultimate laws from which they result, they both fall short of the highest certainty that laws are capable of.

I showed here that laws of causation that are derivative, and compounded of simpler laws, are not only •less general than the simpler laws from which they result, but also •less certain, less entitled to be relied on as universally true. But the certainty-gap between •simpler laws and •the less general laws derived from them, is trifling compared with the certainty-gap between •simpler laws and •uniformities that aren’t known to be laws of causation at all. Until these are resolved, we can’t tell how many collocations as well as laws their truth may depend on; so we can never confidently extend them to cases where we haven’t assured ourselves (by trial) that the required collocations of causes actually exist. The property that philosophers usually regard as characteristic of empirical laws—namely, being unfit to be relied on beyond the limits of time, place, and circumstance in which the observations have been made—really and strictly belongs only to laws in this class. They are ‘empirical laws’ in a stronger and more direct sense; and except where the context plainly indicates otherwise I shall use the phrase ‘empirical laws’ only to refer to uniformities—whether of succession or of coexistence—that aren’t known to be laws of causation.

NOTES

[1] Water, of which eight-ninths in weight is oxygen, dissolves most bodies that contain a high proportion of oxygen. . . .; bodies largely composed of combustible elements. . . .are soluble in bodies of similar composition. . . . This empirical generalisation is far from being universally true; no doubt because it is a remote and therefore easily defeated result of general laws that are too deep for us at present to penetrate; but it will probably in time suggest lines of inquiry that will lead to the discovery of those laws.

Chapter 17. Chance and its elimination

§1. Empirical laws, then, are observed uniformities concerning which the question ‘Are they laws of causation?’ must remain undecided until •they are explained deductively or •some means are found of applying the Method of Difference to the case; and I showed in chapter 16 that until a uniformity can in one of these ways be removed from the class of empirical laws and classified either as a law of causation or a demonstrated results of laws of causation, we can’t be sure that it is true beyond the spatial and other limits within which it has been found true by actual observation. There remains the question: How are we to sure that it is true even within those limits? How much experience is needed for a generalisation that rests solely on the Method of Agreement to be considered sufficiently established, even as an empirical law? Here I explicitly set this question aside, and now it’s time to try to solve it.

We found that the Method of Agreement doesn’t prove •causation, and can therefore only be used for ascertaining •empirical laws. But we also found that it has a second characteristic imperfection, namely tending to make uncertain even conclusions of the sort that it is in itself adapted to prove. That’s because of plurality of causes. Although two or more cases where the phenomenon a has been met with have no common antecedent except A, this doesn’t prove that there is any connection between a and A, because a may have many causes, and may have been produced in these different instances not •by anything that the instances had in common but •by a variety of different elements. But I remarked that as the number of instances pointing to A as the antecedent grows, the uncertainty of the method lessens and the existence of a law connecting A with a comes closer to certainty. Now we have to determine how much experience is needed for this certainty to be regarded as practically attained, and the connection between A and a to be accepted as an empirical law.

In more familiar terms: After how many and what sort of instances are we entitled to conclude that an observed coincidence between two phenomena is not the effect of chance?

For understanding the logic of induction, it is vitally important to have a distinct conception of •what is meant by chance, and of •how the phenomena that common language ascribes to that abstraction—·chance·—are really produced.

§2. Chance is usually spoken of in direct antithesis to law. The thought is that whatever can’t be ascribed to any law is due to chance. But it’s certain that everything that happens is the result of some law; is an effect of causes, and could have been predicted from a knowledge of the existence of those causes and of their laws. When I turn up the Queen of diamonds, that’s a consequence of its place in the pack. Its place in the pack was a consequence of •how the cards were shuffled, or of •the order in which they were played in the last game; and those again were effects of prior causes. At every stage, if we knew enough about the causes in existence, it would have been theoretically possible—·not necessarily possible in practice·—to foretell the effect.

An event occurring by chance may be better described as a coincidence from which we have no basis for inferring a uniformity—the occurrence of a phenomenon in certain circumstances without this giving us reason to think that it will happen again in those circumstances. But this implies that not all the circumstances have been taken into account. Whatever the phenomenon is, since it has occurred once we can be sure that if all the same circumstances were repeated it would occur again; and not only if all—there’s some particular subset of those circumstances on which the phenomenon is invariably consequent. It isn’t connected in any permanent manner with most of the circumstances; its conjunction with those is said to be the effect of chance, to be merely casual. Facts casually conjoined are separately the effects of causes and therefore of laws, but of different causes, and causes not connected by any law.

So it’s wrong to say that any phenomenon is produced by chance; but it is all right to say that two or more phenomena are conjoined by chance, i.e. that they coexist or succeed one another only by chance. That means that there’s no causal relation between them, i.e. it is not the case that they are related

•as cause and effect, or as

•effects of a single cause, or as

•effects of causes that are related by a law of coexistence, or

•effects of a single collocation of primeval causes.

. . . .There is no simple test for this. A coincidence can occur again and again, and yet be only casual. . . . The recurrence of the same coincidence more than once, or even its frequent recurrence, doesn’t prove that it is an instance of any law—doesn’t prove that it is not casual, or (in common language) ‘the effect of chance’.

But when a coincidence can’t be deduced from known laws or proved by experiment to be itself a case of causation, the frequency of its occurrence is the only basis we have for inferring that it is ·not casual but· the result of a law. I’m not talking about its absolute frequency, ·i.e. the answer to the question ‘How often has it occurred?’·. The question is not whether the coincidence occurs often or seldom. . . ., but whether it occurs more often than chance will account for—more often than it would be reasonable to expect if the coincidence were casual. So we have to decide what degree of frequency in a coincidence can be accounted for by chance, and there can be no general answer to this. All I can do is to state the principle by which the answer must be determined; the answer itself will be different in every different case.

Suppose that one of the phenomena, A, exists always, and the other phenomenon, B, exists only occasionally. It follows that every instance of B will be an instance of B’s coincidence with A, and yet the coincidence will be merely casual, not the result of any connection between them. The fixed stars have been in existence ever since the beginning of human experience, and all phenomena that have come under human observation have. . . .coexisted with them; yet this coincidence, though just as invariable as what exists between any one (x) of those phenomena and x’s own cause, doesn’t prove that the stars are in any way causally connected with x. This is as strong a case of coincidence as can possibly exist—much stronger in mere frequency than most of the ones that do prove laws—but it doesn’t here prove a law. Why not? Because the stars exist always and therefore coexist with every other phenomenon, whether connected with it by causation or not. The uniformity is no greater than would occur if there were no such connection.

On the other hand, suppose we’re inquiring whether there’s any connection between rain and some particular ·kind K of· wind. We know that rain occasionally occurs with every wind; so the connection ·between rain and K wind·, if it exists, can’t be an actual law; but still rain may be connected with K wind through causation. They can’t always be effects of a single cause (for if they were they would regularly coexist), but there may be some causes common to them both, so that to the extent that either of them is produced by those common causes they will. . . .be found to coexist. How are we to ascertain this? The obvious answer is: by observing whether rain occurs with K wind more often than with any other. But that’s not enough, for it might be because K wind blows more often than any other; so that its blowing more often in rainy weather is merely what you’d expect if K wind had no connection with the causes of rain. . . . In England, westerly winds blow for about twice as much of the time as do easterly winds. So if it rains only twice as often with a westerly wind as with an easterly one, that’s no reason to infer that any law of nature is at work in the coincidence. If it rains more than twice as often, we can be sure that some law is concerned. Either

•there’s some cause in nature which in this climate tends to produce both rain and a westerly wind, or

•a westerly wind itself has some tendency to produce rain.

But if it rains less than twice as often, we can draw a directly opposite conclusion, inferring that the occurrence of rain is connected •with causes adverse to westerly winds or with •the absence of some cause that produces such winds; and though it may still rain much oftener with a westerly wind than with an easterly, that wouldn’t proving any connection between rain and westerly wind; quite the contrary, it would prove a connection between rain and easterly wind. . . .

So here are two examples:

•one where the greatest possible frequency of coincidence, with no instance to the contrary, doesn’t prove that there is any law; and

•one where a much lower frequency of coincidence (perhaps even lower than the frequency of noncoincidence) does prove that there is a law.

The same principle is at work in both. In both we consider the positive frequency of the phenomena themselves, and on that basis calculate how frequently they would coincide if there were no connection •between them or •between one of them and some cause tending to block the other. If we find a greater frequency of coincidence than this, we conclude that

there’s some connection: one of the phenomena can under some circumstances cause the other, or there’s something capable of causing them both.

And if we find a lesser frequency, we conclude that

there’s some blocking: one of the phenomena, or some cause that produces one of them, can counteract the production of the other.

We have thus to deduct from the observed frequency of coincidence as much as can be the effect of chance, i.e. of the mere frequency of the phenomena themselves; and the remainder—if there is one—is the residual fact that proves the existence of a law.

The frequency of the phenomena can be ascertained only within definite limits of space and time. That’s because it depends on the quantity and distribution of the primeval natural agents, and we can’t know anything about that except by human observation, since we can’t find any law about it enabling us to infer the unknown from the known. But for present purposes this is no disadvantage, because it merely confines the question within the same limits as the data. The coincidences occurred in certain places and times, and within those we can estimate how frequently such coincidences would be produced by chance. If we find from observation that A exists in one case out of every two, and B in one case out of every three, then if there’s neither connection nor opposition between them or between any of their causes, the instances in which A and B will both exist, i.e. will coexist, will be one case in every six. For A exists in three cases out of six; and B—existing in one case out of every three independently of whether A is present or absent—will exist in one case out of those three. Of the six cases, therefore, we can expect there to be

•two in which A exists without B,

•one in which B exists without A,

•two in which neither B nor A exists, and

•one in which they both exist.

If we find that A and B coexist oftener than in one case out of six,. . . .there is some cause in existence that tends to produce a conjunction between A and B.

Generalising this result, we can say that if A occurs in a larger proportion of the B cases than of the not-B cases, then B will also occur in a larger proportion of the A cases than of the not-A cases, and there’s some causal connection between A and B. If we could track back to the causes of A and B, we would find somewhere along the line some cause or causes common to both; and if we could ascertain what these are we could form a generalisation that would be true without restriction of place or time. But until we can do that, the fact of a connection between A and B remains an empirical law.

§3. Having considered how it can be determined whether any given conjunction of phenomena is •casual or •the result of some law, we need now to complete the theory of chance by considering the effects that are partly the result of chance and partly of law, i.e. cases where •the effects of casual conjunctions of causes are habitually blended in one result with •the effects of a constant cause.

This is composition of causes, with a special feature: instead of two or more causes intermixing their effects in a regular manner, we now have one constant cause producing an effect that is successively modified by a series of variable causes. As summer advances, the sun’s approach towards a vertical position tends to produce a constant •increase of temperature; but •this effect of a constant cause is mixed with the effects of many variable causes—winds, clouds, evaporation, lightning and the like—so that the temperature on any given day depends partly on these fleeting causes and only partly on the constant cause. If the effect of the constant cause is always accompanied and disguised by effects of variable causes, it’s impossible to ascertain the law of the constant cause in the ordinary way by observing it apart from all other causes. That creates a need for an additional rule of experimental inquiry.

When the action of a cause A is liable to be interfered with. . . .by different causes at different times, and when these are so frequent or so indeterminate that we can’t exclude them all from any experiment, though we can vary them, we can try to discover what the effect is of all the variable causes taken together. This is how we do it:

•We make as many trials as possible, keeping A invariable and varying everything else as much as possible. The results of these different trials will naturally be different, because their indeterminate modifying causes are different. If we find that these results oscillate about a certain point—one experiment giving a result a little greater, another a little less; one giving a result tending a little more in one direction, another a little more in the opposite direction—while the average or mid-point doesn’t vary. . . ., then that mean or average result is the part in each experiment that is due to the cause A, and is the effect that would have occurred if A had acted alone; the variable remainder is the effect of chance, i.e. of causes whose coexistence with A was merely casual.

This induction counts as sufficient if any increase in the number of trials doesn’t materially [see Glossary] alter the average.

This kind of elimination, in which what we eliminate is not one assignable cause but a multitude of floating unassignable ones, can be called ‘the elimination of chance’. We produce an example of it when we repeat an experiment ·several times· so as to get rid of the effects of the unavoidable errors of each individual experiment by taking the mean of the different results. When there’s no permanent cause that would produce a tendency to error in one direction, we are justified by experience in assuming that the errors on one side will—in a certain number of experiments—just about balance the errors on the opposite side. So we go on repeating the experiment until any change in the over-all average falls within limits of experimental error. How those limits are set will depend on what we are aiming to discover by our inquiry.[1]

§4. I have been assuming that the effect of the constant cause A is such a large and conspicuous part of the over-all result that there’s no room for doubt that it exists, and the eliminating process is merely an attempt to ascertain how much of the over-all result is caused by A, i.e. to discover what A’s exact law is. But in some cases the effect of a constant cause is such a small portion of the total upshot that it escapes notice; and the fact that there is an effect arising from a constant cause is first learned by the process whose usual role is only to ascertain the quantity of that effect. This happens in cases where a given effect E is •known to be chiefly determined by changeable causes and •not known not to be wholly so determined. In that situation we run a large number of trials, watching to see if we get either of these results:

(i) The effects of the different changeable causes cancel one another out, ·and the series homes in on E·.

(ii) The long series of trials homes in on a definite result, but it isn’t precisely E; it differs from E by an amount that is small in comparison with the total effect, but it is definitely there in our results.

In case (i) we conclude that the changeable causes are the only cause of E; in (ii) we conclude that some constant cause is at work, making the results of our trials oscillate around a definite point that isn’t quite E, and we may hope to discover what that cause is by some of the methods I have presented. This last process can be called the discovery of a residual phenomenon by eliminating the effects of chance.

That is how loaded dice can be discovered. Of course no die is so clumsily loaded that it always comes up with the same number; if that happened, the fraud would be instantly detected. The loading, a constant cause, mingles with the changeable causes that determine how the die is thrown in each individual instance. If the die wasn’t loaded and the throw depended entirely on those changeable causes, in a long enough series of throws the changeable causes would balance one another so that the numbers on the die would come up about the same number of times. If we throw the die often enough so that we stop having any material effect on the relative frequencies of the numbers, and find that the stable distribution of numbers that we eventually reach has one number coming up significantly more often than any other, we can be sure that some constant cause is at work favouring that number—i.e. that the die is not fair—and we know exactly how unfair it is. . . .

§5. After these general remarks about the nature of chance, I’m now ready to consider how we can become sure that a conjunction between two phenomena that has been observed a certain number of times is not casual, but a result of causation. When we are sure of that we can accept this going-together of the two phenomena as one of the uniformities of nature, though (until accounted for deductively) only as an empirical law.

Suppose that the phenomenon B has never been observed except in conjunction with A. Even then, the probability that they are connected isn’t measured by the total number of instances in which they have been found together, but by that number minus the number stating the absolute frequency of A. If A exists always, and therefore coexists with everything, no number of instances of A together with B would prove a connection—as in the example of the fixed stars. If A occurs so commonly that we can presume it to be present in half of all the cases that occur, and therefore in half the cases in which B occurs, our only evidence that there’s a connection between A and B would have to come from A’s occurring in more than half the occurrences of B.

In addition to the question

(i) In a great multitude of trials, how many coincidences can on average be expected to arise from chance alone?

there is also the question

(ii) In a number of instances smaller than that required for striking a fair average, how much deviation from that average can be expected from chance alone?

That is, we have to consider not only (i) the general result of the chances in the long run, but also (ii) what the extreme limits of variation from the general result are that can occasionally be expected as the result of some smaller number of instances. The consideration of (ii) and any further consideration of (i) belong to what mathematicians term the doctrine of chances, or in a grander phrase, the Theory of Probabilities.

NOTES

[1] I have been speaking of the mean as if it were the average. But for purposes of inductive inquiry the mean is not the average or arithmetical mean, though the difference can be disregarded in informal illustrations of the theory. If the deviations on one side of the average are much more numerous than those on the other (these last being fewer but bigger), the effect due to the invariable cause won’t coincide with the average but will be either below or above it, the deviation being toward the side on which the greatest number of the instances are found. . . .

Chapter 18. The calculation of chances

§1. In his Essai philosophique sur les probabilités, Laplace wrote:

‘Probability has reference partly to our ignorance, partly to our knowledge. We know that among three or more events exactly one must happen, but we have no grounds on which to pick just one and believe that it will happen. In this state of indecision, we can’t say with certainty anything about which one will occur. But we can say of each of them that it probably won’t occur, because we know of several equally possible events that exclude its occurrence, and only one that favours it.

‘The theory of chances consists in •reducing all events of the same kind to a certain number of equally possible cases, i.e. cases that we are equally undecided about; and •determining the number of these cases that are favourable to the event whose probability we are looking for. The ratio of that number to the number of all the possible cases is the measure of the probability. So the probability ·of an event· is a fraction, having for its numerator the number of cases favourable to the event and for its denominator the number of all the possible cases.’

For a calculation of chances, then, according to Laplace, two things are necessary: •we must know that of several events some one and only one will certainly happen; and •it mustn’t be the case that we know, or have any reason to expect, that this or that one in particular is going to happen. It has been contended that these aren’t the only requirements, and that Laplace has overlooked a necessary part of the foundation of the doctrine of chances. To declare two events to be equally probable (say these critics) we need three things:

•to know that one of the two must happen,

•to not know which one will happen, and

•to know from experience that the two events occur equally often.

Why when we flip a coin do we think it to be equally probable that it will come up heads or tails? Because we know that in any large number of throws, heads and tails are thrown about equally often; and that the more throws we make the nearer we come to perfect equality. We can if we wish know this •by actual experiment, or •by the daily experience that life gives us of events of the same general sort, or •deductively from the effect of mechanical laws on a symmetrical body acted on by forces varying indefinitely in quantity and direction. We may know it, in short, either by specific experience or on the evidence of our general knowledge of nature. But we must know it somehow if we are to call the two events equally probable; and if we don’t know it, we’re running as much risk in staking equal sums on the result as in laying odds.

That’s the view of the subject that I took in the first edition of the present work; but I have since become convinced that the theory of chances, as conceived by Laplace and by mathematicians generally, doesn’t have the basic fallacy of which I had accused it.

Remember that an event’s probability is not a quality of the event itself; it’s merely a name for the strength of the grounds that we or others have for expecting it. The probability of an event to you is a different thing from its the probability to me, and also different from the probability to you after you have acquired additional evidence. The probability to me that John Doe, of whom I know nothing but his name, will die within the year is totally altered when I’m told that he has severe tuberculosis. Yet this knowledge of mine makes no difference to the event itself or to any of the causes on which it depends. Every event is in itself •certain, not •probable; if we knew all ·the relevant facts· we would either know that it will happen or know that it won’t. Its probability to us is just the degree of expectation of its occurrence that our present evidence entitles us to have.

Bearing this in mind, I think it must be admitted that even when we have no knowledge to guide our expectations except the knowledge that what happens must be some one of a certain number of possibilities, we can still reasonably judge that one supposition is more probable to us than another; and if the outcome matters to us in any way, we ought to act on that judgment.

§2. Suppose we’re required to take a ball from a box of which we know only that it contains black balls and white ones, and none of any other colour. We know that the ball we select will be either black or white, but we have no basis for expecting one colour rather than the other. In that case, if we have to make a choice and bet something on one or the other supposition, we’ll have no prudential reason to select either colour, and we’ll act precisely as we would have acted if we had known that the box contained an equal number of black and white balls. But though our conduct would be the same, it wouldn’t be based on a guess that the balls were in fact equally divided. To see why, suppose we •know for sure that the box contains 99 balls of one colour and only one of the other, but •don’t know which colour has only one and which has 99; in that case the drawing of a white and of a black ball will be equally probable to us. We’ll have no reason for staking anything on one event rather than on the other; the option between the two will be a matter of indifference; in other words, it will be an even chance.

Now vary the case: suppose that instead of two there are three colours—white, black, and red—and that we’re entirely ignorant about how many of each. We have no reason to expect one more than another, and if we have to bet we’ll regard each colour as on a par with each of the other two. But if there’s a question of betting on (say) white as against red or black, would it be a matter of indifference which way we betted? Surely not! Because black and red are each as probable to us as white, the two together must be twice as probable. We would in this case expect not-white rather than white, and so much ‘rather’ that we would bet two to one on it. It’s true that for all we know there may be more white balls than black and red together; and in that case our bet would, if we knew more, be seen to be a disadvantageous one. But then for all we know to the contrary there may be more red balls than black and white, or more black balls than white and red, and in that case the effect knowing more would be to prove to us that our bet was more advantageous than we had supposed it to be. But in the actual state of our knowledge there’s a rational probability of two to one against white–a probability fit to be made a basis of conduct. No reasonable person would lay an even wager in favour of white against black and red; though against black alone or red alone he might do so without imprudence.

So the common theory of the calculation of chances seems to be tenable. Even when we know nothing except the number of the possible and mutually excluding contingencies, and are entirely ignorant of their comparative frequency, we may have grounds—ones that can be evaluated numerically—for acting on one supposition rather than on another; and this is the meaning of probability.

§3. The reasoning here is based on the obvious principle that when there are several mutually exclusive kinds of possible outcome, it’s impossible for each of those kinds to be a majority of the whole. On the contrary, there must be a majority against each kind except one at most; and if any kind has more than its share in proportion to the total number, the others collectively must have less. Granting this axiom, and assuming that we have no ground for selecting any one kind as more likely than the rest to surpass the average proportion, it follows that we can’t rationally presume this in our betting. Thus, even in this extreme case of the calculation of probabilities—which doesn’t rest on special experience at all—the logical ground of the process is our knowledge of the laws governing the frequency of occurrence of the different cases. But this is knowledge of universal and axiomatic truths, and doesn’t bring in any specific experience or any considerations arising from the special nature of the problem under discussion.

But I can’t conceive of a case where we ought to be satisfied with an estimate of chances based on the absolute minimum of knowledge concerning the subject—except in games of chance, the purpose of which requires ignorance instead of knowledge. It’s clear that in the case of the coloured balls a very slight ground for thinking that the white balls outnumbered each of the other colours would undermine the whole calculation made in our previous state of ·ignorance and· indifference. It would equip us with more advanced knowledge, in which the probabilities-to-us were different from what they had been before; and in estimating these new probabilities we would have to proceed on a totally different set of data, provided by specific knowledge of facts rather than by mere counting of possible suppositions. We ought always to try to get such additional data, and it’s •always possible to get some that are, if not good bases for action, at least better than none at all; well, •always unless we’re dealing with something that is equally beyond the range of our means of knowledge and our practical uses.[1]

It is obvious, too, that even when the probabilities are derived from observation and experiment,

•a very slight improvement in the data, by better observations or by attending more fully to the special circumstances of the case

is more useful than

•the most elaborate application of the calculus to probabilities based on the unimproved data.

The neglect of this obvious point has led to misapplications of the calculus of probabilities that have made it the scandal of mathematics. Just look at how it is applied to (i) the credibility of witnesses and to (ii) the correctness of the verdicts of juries. (i) Common sense would dictate that it is impossible to say what the average level of truthfulness etc. is for mankind in general or for any class of people ·or indeed for any individual·; and even if this were possible, the use of it for such a purpose—·i.e. for deciding how much trust to place in the testimony of witness John Doe·—reveals a misunderstanding of the use of averages. . . . In the case of a witness, persons of common sense will go by the degree of consistency of his statements, his conduct under cross-examination, and the relation of the case itself to his interests, his partialities, and his mental capacity, instead of applying such a rough standard (even if it could be verified) as the ratio of true to erroneous statements that he had made in the course of his life.

(ii) Some mathematicians have set out from the proposition that the judgment of any one judge or juryman is at least somewhat more likely to be right than wrong, and have inferred from this that the chance of a number of persons all reaching the same wrong verdict is small in proportion that the number of judges or jurymen is large; so that if there are enough judges the correctness of their judgment can be raised almost to certainty. This raises the question of the effect on the moral position of the judges by multiplying their numbers—the virtual destruction of their individual responsibility, and the weakening of their mental focus on the subject—but let all that pass. I merely point out the fallacy of reasoning from •a wide average to •cases necessarily differing greatly from any average. If we look at all the legal cases judge J has been involved in, perhaps he has more often been right than wrong; but now look at his record in all the complicated and otherwise tricky cases he has been involved in, it’s likely enough that in them he has more often been wrong than right. (Why focus on the difficult cases? Because it’s only in them that it matters much who the judges are.) And there’s another point: if judge J’s errors in tricky cases have arisen from the intricacy of the case or from some common prejudice or mental infirmity, the odds are that such factors will have acted on most of the other judges in the same way; so that increasing the number of judges will increase the probability of a wrong decision.

These are merely samples of the errors often committed by men who, having learned to use difficult algebraic formulae in estimating chances in complex cases, would rather •use those formulae to compute what the probabilities are to a person who is half-informed about a case than •look for ways of being better informed. If we’re to get anything scientifically useful out of the doctrine of chances, we must first lay a foundation for an evaluation of the chances by getting as much ·relevant· factual knowledge as we can. The knowledge required is that of the comparative frequency with which the different events actually occur. For the purposes of the present work, therefore, it is permissible to suppose that conclusions about the probability of a fact of kind K rest on our knowledge of the proportion between •the cases where K facts occur and •those in which they don’t occur; this knowledge being either derived from specific experiments or deduced from our knowledge of causes that tend to produce K facts compared with causes that tend to prevent them.

Such calculation of chances is based on an induction, and the calculation isn’t legitimate unless the induction is valid. It’s not stopped from being an induction by the fact that it doesn’t prove that a K event occurs in all cases of sort S but only that out of a given number of S cases a K event occurs in about such-and-such a number. The fraction that mathematicians use to designate the probability of an event is the ratio of these two numbers; the ascertained proportion between •the number of cases in which a K event occurs and •the number of all the cases (i.e. those in which a K event does occur plus those in which it doesn’t). In playing at coin-tossing, the S cases are throws of the coin, and the probability of heads is one-half because if we throw often enough heads is thrown about half the time. In the cast of a die, the probability of 6 is one-sixth; not simply

•because there are six possible outcomes of which 6 is one, and we know no reason why one should turn up rather than another,

—though I have accepted the validity of this ground if it were the best we could do—but

•because we do actually know, either by reasoning or by experience, that in a hundred or a million throws 6 is thrown in about one-sixth of that number.

§4. When I say ‘either by reasoning or by experience’ I mean specific experience. When we are estimating probabilities it makes a difference which of these two is the basis for our assurance. The probability of events •as calculated from their mere frequency in past experience is a less secure basis for practical guidance than their probability •as deduced from an equally accurate knowledge of the frequency of occurrence of their causes.

The generalisation that an E event occurs in ten out of every hundred S cases is as real an induction as if the generalisation were that it occurs in all S cases. But when we reach this conclusion by merely counting S instances in actual experience and comparing the number of them in which an E has occurred with the number in which it hasn’t, our evidence is only that of the Method of Agreement, and the conclusion amounts to a mere empirical law. We take a step beyond this when we •ascend to the causes on which the occurrence or non-occurrence of E events depends, and •form an estimate of the comparative frequency—·among all S cases·—of the causes favourable to E and of those unfavourable to E. These are data of a higher order, by which the merely empirical law. . . .will be either corrected or confirmed, and either way we’ll get a more correct measure of probability than is given by the numerical comparison underlying the empirical law. A writer in the Prospective Review recently said, rightly, that in the kind of examples by which the doctrine of chances is usually illustrated—namely, balls in a box—the estimate of probabilities is supported by reasons of causation, which are stronger than reasons from specific experience. ‘What is the reason that in a box where there are nine black balls and one white, we expect to draw a black ball nine times as much (i.e. nine times as often, frequency being the gauge of intensity in expectation) as a white? Obviously because the local conditions are nine times as favourable; because the hand may alight in nine places and get a black ball, while it can only alight in one place and find a white ball; like the reason why we don’t expect to succeed in finding a friend in a crowd, because the conditions for our coming together are so many and so difficult. This wouldn’t hold to the same extent if the white ball were larger than the black ones, and in that case the probability would be different.’

It is in fact obvious that once causation has been admitted as a universal law, that law becomes the only rational basis for our expectation of events. For someone who recognises that every event depends on causes, a thing’s having happened once is a reason for expecting it to happen again only because it happening once shows that there is—or is liable to be—a cause adequate to produce it. The frequency of the particular event, apart from any thought of its cause, can’t give rise to any induction except an inductio per enumerationem simplicem [see Glossary]; and the precarious conclusions reached in this way are superseded, and disappear from the ·battle·field as soon as the principle of causation shows up there.[2]

Still, although an estimate of probability based on causes is theoretically better, in practice it can’t be done much. In almost all cases where chances can be estimated precisely enough to be of any practical value, the numerical data are drawn not from •knowledge of the causes but from •experience of the events themselves—

•the probabilities of life at different ages or in different climates;

•the probabilities of recovery from a particular disease;

•the chances of the birth of male or of a female offspring;

•the chances of the destruction of houses by fire;

•the chances of the loss of a ship in a particular voyage;

—these are all deduced from mortality statistics, hospital records, registers of births, registers of shipwrecks, and so on, i.e. from the observed frequency not of the causes but of the effects. We go about it in this way because in all these contexts the causes are either not open to direct observation at all, or not with the required precision, and we have no way of judging of their frequency except from the empirical law provided by the frequency of the effects. But the inference still entirely depends on causation alone: we reason from an effect to a similar effect by passing through the cause. If the actuary in an insurance office infers from his tables that among a hundred 50-year-old persons now living five on average will reach the age of seventy, his inference is legitimate; not for the simple reason that this is the proportion who have reached seventy in the past, but because that statistical fact shows that 5:95 is the proportion existing at that place and time between the causes that prolong life to the age of seventy and the causes tending to bring it to an earlier close.[3]

§5. From the preceding principles it’s easy to work out how to demonstrate the theorem that is the basis for the use of the concept of probability in application to •the occurrence of a given event or •the reality of an individual fact. The signs or evidences by which a fact is usually proved are some of its consequences; and the inquiry hinges upon determining what cause is most likely to have produced a given effect. The theorem applicable to such investigations is the Sixth Principle in Laplace’s Essai philosophique sur les Probabilités, which he describes as the ‘fundamental principle of the branch of the Analysis of Chances that consists in ascending from events to their causes.’[4]

Given an effect to be accounted for, with several causes that might have produced it, though nothing is known about their role (if any) in this particular case, the probability that the effect was produced by any one of these causes is •the antecedent probability of that cause multiplied by •the probability that that cause, if it existed, would produce the given effect.

Let E be the effect and C_1 and C_2 the two causes by either of which E might have been produced. To find the probability that it was produced by C_1 (say), ascertain which of the two is more likely to have existed, and which of them, if it did exist, was more likely to produce the effect E: the probability sought is a compound of these two probabilities.

[Mill has slipped here. He speaks of ‘these two probabilities’, but he hasn’t mentioned two probabilities, only two probability-comparisons.]

CASE I: The causes are alike in the second respect; C_1 and C_2 when they exist are equally likely (or equally certain) to produce E; but C_1 is twice as likely as C_2 to exist, i.e. is twice as frequent a phenomenon. Then it is twice as likely to have existed in this case, and to have been the cause that produced E.

Explanation: C_1 exists in nature twice as often as C_2, so in any 300 cases in which one or other existed, C_1 has existed 200 times and C_2 100. But either C_1 or C_2 must have existed wherever E is produced; therefore, in 300 times that E is produced, C_1 was the producing cause 200 times, C_2 only 100; i.e. in the ratio of 2 to 1. Thus, if the causes are alike in their ability to produce the effect, the probability as to which actually produced it is in the ratio of their prior probabilities.

CASE II: The causes are equally frequent, i.e. equally likely to have existed, but not equally likely if they did exist to produce E. Specifically, in three times in which C_1 occurs it produces that effect twice, while C_2 in three occurrences produces it only once. Since the two causes occur with equal frequency, in every six times that either one or the other exists, C_1 exists three times and C_2 three times. C_1 produces E in two of its three occurrences, while C_2 produces E once in its three occurrences. Thus, in the whole six times, E is produced only three times; but of those three it is produced twice by C_1 and only once by C_2. Consequently, when the antecedent probabilities of the causes are equal, the chances that the effect was produced by them are in the ratio of the probabilities that if they did exist they would produce the effect.

CASE III: The causes are unalike in both respects. This is solved by the solutions of Cases I and II. For, when a quantity depends on two other quantities in such a way that while either of them remains constant it is proportional to the other, it must be proportional to the product of the two quantities, product being the only function of the two that obeys that law of variation. Therefore, the probability that E was produced by either cause is the antecedent probability of the cause’s existing multiplied by the probability that if it existed it would produce E. QED.

Explanation: Let C_1 occur twice as often as C_2; and let C_1 produce E twice in four occurrences, and C_2 produce E three times in four occurrences. C_1’s antecedent probability is to C_2’s as 2 to 1; the probabilities of their producing E are as 2 to 3; the product of these ratios is the ratio of 4 to 3; and this will be the ratio of the probabilities that C_1 or C_2 was the producing cause in the given instance. Since C_1 is twice as frequent as C_2, out of twelve cases in which one or other exists, C_1 exists in 8 and C_2 in 4. But out of its eight occurrences C_1 produces E in only 4, while C_2 out of its four cases produces E in 3. So E is produced at all in seven of the twelve cases; in four of these it is produced by C_1, in three by C_2; hence the probabilities of its being produced by C_1 and by C_2 are as 4 to 3, and are expressed by the fractions 4⁄7 and 3⁄7. QED.

§6. How does the doctrine of chances relate to the special problem I discussed in chapter 17? I mean the problem of how to distinguish coincidences that are casual from ones that are the result of law, i.e. from ones in which the facts that accompany or follow one another are somehow connected through causation.

The doctrine of chances provides means by which, if we know the average number of coincidences to expect between two phenomena connected only casually, we can calculate how often any given deviation from that average will occur by chance. If the probability of any casual coincidence is 1/m, the probability that the same coincidence will be repeated n times in succession is 1/(m × n). In one throw of a die the probability of 4 is 1/6; so the probability of throwing 4 twice in succession is 1/6^2 = 1/36. To see why, consider: 4 is thrown once in six throws, or six in thirty-six throws; and of those six, when die is cast again 4 will be thrown only once; making once in thirty-six throws altogether. The chance of throwing 4 three times in succession is 1/6^3, which is 1/216. . . .

So we have a rule by which to estimate the probability that any given series of coincidences [see Glossary] arises from chance, provided we know the probability of a single coincidence. If we can get an equally precise expression for the probability that the same series of coincidences arises from causation, we’ll only have to compare the numbers. But we usually can’t do this. Let us see how near we can come, in practice, to the necessary precision.

The question falls within ·the scope of· Laplace’s sixth principle, which I have just demonstrated. The series of coincidences may have originated either in (i) a casual conjunction of causes or in (ii) a law of nature. The probability that the series originated in manner (i) is given by •the antecedent probability of its being the case multiplied by •the probability that if it were the case it would produce that series of coincidences; and similarly, mutatis mutandis, for the probability that the series came from (ii) a law of nature. Well, the two are on a par as regards probability-ofproducing-the-effect: if either of them were real, that series of coincidences would certainly occur. So the probability that the coincidences are produced by this or that one of the two causes is the antecedent probability of that cause’s existing. The antecedent probability of (i) is a quantity we can measure. How exactly we can estimate the antecedent probability of (ii) will vary according to the nature of the case.

In some cases if the coincidence is result of causation we know what the cause must be—e.g. we know that if a consecutive series of 4s isn’t accidental it must arise from the loading of the die. In such a case we may have a basis, in the characters of the parties concerned or other such evidence, for a conjecture as to the antecedent probability of such an event; but we can’t possibly estimate that probability with anything like numerical precision. But the counterprobability—i.e. the probability that a consecutive series of 4s is accidental—dwindles very fast as the series continues; so that we soon reach the stage at which the chance that the die has been loaded, however small in itself, must be greater than the chance of a casual coincidence; and on this basis a practical decision can generally be reached without much hesitation if it’s possible to repeat the experiment.

But when the situation is like the one we were looking at in chapter 17—i.e. when the coincidence can’t be accounted for by any known cause, so that if the connection between the two phenomena is causal it must be the result of some law of nature that we don’t yet know—then ·we have a new problem·. We may be able to estimate the probability of a casual coincidence, but the probability of the counter-supposition, namely the existence of an undiscovered law of nature, is clearly something we can’t estimate even approximately. To have a basis for such an estimate we would need to know

•what proportion of all the individual sequences or coexistences occurring in nature are the result of law, and •what proportion are mere casual coincidences.

Obviously, we can’t make any plausible conjecture about this proportion, much less assign it a number; so we can’t attempt any precise estimation of the comparative probabilities. But we are sure of this much: the detection of an unknown law of nature—of some previously unrecognised constancy of conjunction among phenomena—is not an uncommon event. Therefore, if

•the number no of instances in which a coincidence is observed is so much larger than •the number nc that would occur on the average from chance that it would be an extremely uncommon event for no coincidences to occur from accident alone,

then we have reason to conclude that the coincidence is an effect of causation and can be accepted (subject to correction from further experience) as an empirical law. We can’t pin it down more precisely than this, but in most cases this level of precision is all we need to resolve any practical doubt.

[This was originally a footnote.] For a fuller treatment of many interesting questions in the theory of probabilities I recommend John Venn’s recent The Logic of Chance, which is one of the most thoughtful and philosophical works on any subject connected with logic and evidence that I know of. Some criticisms ·of my work· contained in it have helped me to revise the corresponding chapters of the present work. Any reader of Venn’s work who is also a reader of this will see which of his opinions I don’t accept.

NOTES

[1] It seems to me, indeed, that the calculation of chances in the absence of data based on special experience or on special inference must in the vast majority of cases break down, from sheer impossibility of assigning any principle by which to be guided in setting out the list of possibilities. In the case of the coloured balls we can easily list the possibilities because we determine what they are. But now take a case that is more like the ones that occur in nature: instead of three colours, let the box contain all possible colours, and suppose that we are ignorant of the comparative frequency with which different colours occur in nature or in the productions of art. How are we to list the possibilities? Is every distinct shade to count as a colour? If so, is the test ·of distinctness· to be ·conducted by· a common eye or an educated one—a painter’s, for instance? Answers to these questions could make the difference between whether the chances against some particular colour should be estimated at 10:1 or 20:1 or perhaps 500:1. Whereas if we knew from experience that the particular colour occurs on an average of (say) 33 times in every hundred or thousand, we wouldn’t need to know anything about the frequency of the other possibilities or even about how many of them there are.

[2] [This footnote which Mill tied to ‘. . . a cause adequate to produce it’ a few lines back, is a quotation from the Prospective Review article mentioned in the preceding paragraph.] ‘Why do we feel so much more probability added by •the first instance than by •any single subsequent instance? It has to be because the first instance gives us its possibility (a cause adequate to it), while every other only gives us the frequency of its conditions. If no reference to a cause were implied, possibility would have no meaning; yet it’s clear that before the event happened we might have thought it to be impossible, i.e. have believed that there was no physical energy really existing in the world equal to producing it. . . The first time of happening, then, is more important to the whole probability than any other one instance (because it proves the possibility); after that, the number of times becomes important as a sign of the intensity or extent of the cause, and its independence of any particular time. Suppose we want to estimate the probability someone’s being able to perform a tremendous leap a certain number of times; at first we don’t know whether the leap is possible, but the all-important first leap gets rid of that doubt. Every leap after that shows the power to be •more perfectly under ·the athlete’s· control, •greater, and •more invariable, and so it increases the probability. No-one would think of reasoning in this case directly from one instance to the next, without referring to the physical energy that each leap indicated. So it’s clear that we do not ever conclude directly from the happening of an event to the probability of its happening again; rather, we refer to the cause, regarding the past cases as a sign of the cause, and the cause as our guide to the future.’ [Mill interrupts this by suggesting that ‘. . . we do not ever. . . ’ should be ‘. . . we do not in an advanced state of our knowledge. . . ’.]

[3] The writer last quoted says that estimating chances by comparing the number of cases in which the event occurs with the number in which it doesn’t ‘would generally be wholly erroneous’ and ‘is not the true theory of probability’. Well, it’s the theory that forms the foundation of insurance, and of all the calculations of chances in the business of life. The writer’s reason for rejecting the theory is that it ‘would regard as certain an event that had never failed up to now; which is very far from the truth, even for a very large number of constant successes.’ This isn’t a defect in a particular theory, but in any theory of chances. No principle of evaluation can deal with a case such as this writer supposes. If an event has never once failed in a long enough series of trials to eliminate chance, it has all the certainty that an empirical law can provide; it is certain for as long as the relevant collocation of causes continues. If it ever fails, it will be because of some change in that collocation. Now, no theory of chances will enable us to infer the future probability of an event from the past, if the relevant causes have undergone a change.

[4] Laplace doesn’t state the theorem in exactly they way I have stated it, but it’s easy to demonstrate that the two formulations are equivalent.

Chapter 19. Extending derivative laws to adjacent cases

§1. I have frequently remarked that derivative laws are less general than the ultimate laws they are derived from, and also less certain. This is most conspicuous in the uniformities of coexistence and sequence between effects that depend ultimately on different basic causes. Such uniformities always reflect the same collocation of those primeval causes—·i.e. the causes coexist if the effects do, and occur in sequence if the effects do·. If the collocation of the causes varies, though the laws of the causes remain the same, the set of derivative uniformities can and usually will be totally different.

Even where the derivative uniformity is between different effects of a single cause, it won’t exist as universally as the law of the cause does. If a and b accompany or succeed one another as effects of the cause C_1, it doesn’t follow that C_1 is the only cause that can produce them, or that if there’s another cause C_2 that can produce a it must produce b likewise. So it may be that the conjunction a and b doesn’t hold universally, but holds only in the instances in which a arises from C_1. When it is produced by some other cause, a and b may be separated. Day is always in our experience followed by night; but day isn’t the cause of night; both are successive effects of a common cause, the spectator’s periodical move into and out of the earth’s shadow, resulting from •the earth’s rotation and •the illuminating power of the sun. So if day is ever produced by a different set of causes from this, day may not be followed by night. On the sun’s own surface, for instance, this may be the case.

Finally, even when the derivative uniformity is itself a law of causation (resulting from the combination of several causes), it isn’t entirely independent of collocations. If a cause intrudes that wholly or partially counteracts the effect of any one of the combined causes, the effect will no longer conform to the derivative law. Thus, while each ultimate law is vulnerable to frustration from one set of counteracting causes, the derivative law is vulnerable to it from several. And the possibility of the occurrence of counteracting causes that don’t arise from any of the conditions involved in the law itself depends on the original collocations.

It is true that laws of causation, whether ultimate or derivative, are in most cases fulfilled even when counteracted (I said this here)—the cause produces its effect though that effect is destroyed by something else. So the fact that the effect can be frustrated doesn’t harm the universality of the law governing the cause. But it is fatal to the universality of the sequences or coexistences of effects that are the subject-matter of most of the derivative laws flowing from laws of causation. . . . Here’s an example.

From the combination of •a single sun with •an opaque body’s rotation around its axis there results •an alternation of day and night on the whole surface of that opaque body. If one of the combined causes were counteracted—the rotation stopped, the sun extinguished, or a second sun added—this wouldn’t affect the truth of that particular law of causation; it would be still true that one sun shining on an opaque revolving body will alternately produce day and night; but. . . .the derivative uniformity, the succession of day and night on the given planet, would no longer hold.

So the derivative uniformities that aren’t laws of causation always depend to some extent on collocations; and that exposes them to the characteristic infirmity of empirical laws—namely, being acceptable only where the collocations are known by experience to be required for the truth of the law, i.e. only within the conditions of time and place confirmed by actual observation. (I said ‘always’; it should have been ‘always except in the rare case where they depend on one cause rather than a combination of causes’.)

§2. This principle, when stated in general terms, seems clear and indisputable; yet many of the ordinary judgments of mankind—ones that no-one challenges as improper—seem to be inconsistent with it. On what grounds, it may be asked, do we expect that the sun will rise tomorrow? The time-span through which we have made observations includes thousands of past years, but it doesn’t include the future. Yet we infer with confidence that the sun will rise tomorrow, and nobody doubts that we’re entitled to do so. Let us consider what is the basis for this confidence.

In the example in question, we know the causes that the derivative uniformity depends on. They are •the sun giving out light and •the earth rotating and intercepting light. Given a completed induction showing these to be real causes, and not merely. . . .effects of a common cause, the only circumstances that could defeat the derivative law are ones that destroy or counteract one of the combined causes. For as long as the causes exist and aren’t counteracted, the effect will continue. If they exist and aren’t counteracted tomorrow, the sun will rise tomorrow.

Since the causes will exist until something destroys them, everything depends on the probabilities of their being destroyed or counteracted. We know by observation. . . .that these phenomena have continued for (let’s say) 5,000 years. Within that time no cause has appreciably weakened them or counteracted their effect. So the chance that the sun won’t rise tomorrow amounts to the chance that some cause that hasn’t shown up in the smallest degree during 5,000 years will exist tomorrow with enough intensity to destroy the sun, the earth, the sun’s light, or the earth’s rotation, or to produce an immense disturbance in the effect resulting from those causes.

If such a cause will exist tomorrow or at any future time, some cause of that cause must exist now and must have existed during the whole 5,000 years. So if the sun doesn’t rise tomorrow, that will be because there is some cause whose effects

•have through 5,000 years been too small to be perceptible, but

•will overnight become overwhelming.

Since this cause hasn’t been recognised during all those years by observers on our earth, if it’s a single cause it must either •be one whose effects develop gradually and very slowly or •one that existed in regions beyond our observation and is now on the point of arriving in our part of the universe. Now, all causes that we have experience of act according to laws incompatible with the supposition that their effects could be imperceptible for 5,000 years and then swell to immensity in a single day. No mathematical law of proportion between an effect and the quantity or relations of its cause could produce such contradictory results. The sudden development of an effect of which there was no previous trace always arises from the coming together of several distinct causes that haven’t previously been conjoined; but if such a sudden conjunction is going to take place tonight, the causes (or their causes) must have existed during the entire 5,000 years; and their not having once come together during all that time shows how rare that particular combination is. So we have a rigid induction to support us in thinking that the known conditions required for the sun’s rising will exist tomorrow is probable in a degree that can’t be distinguished from certainty.

§3. But this extension of derivative (not causative) laws beyond the limits of observation can only be to adjacent cases. If instead of ‘tomorrow’ I had said ‘twenty thousand years from today’, the inductions would have been anything but conclusive. ·That is, it’s not out of the question that in that stretch of time something might happen to stop the sun from rising·. Consider:

A cause that has, in opposition to very powerful causes, produced no perceptible effect during a considerable period will produce a very considerable effect by the end of a further much longer period.

Nothing in that conflicts with our experience of causes. ·There are at least three ways it could happen·:

(1) An agent whose effect over the past 5,000 years •hasn’t amounted to a perceptible quantity •becomes considerable by accumulating over the next 20,000 years.

(2) There is moving towards us some ·heavenly· body that •hasn’t influenced us during ·the past· 5,000 years but •will get close enough to produce extraordinary effects on us in ·the next· 20,000 years.

(3) Sunrise could be prevented by a certain combination of causes; and although that combination hasn’t arisen in the past 5,000 years it will arise in the next 20,000 years.

So the inductions that authorise us to expect future events grow weaker and weaker the further we look into the future, until eventually they have no significant force.

I have considered the probabilities of the sun’s rising tomorrow, as derived from the real laws; i.e. from the laws of the causes on which the day-night uniformity depends. Let us now see what the situation would be if for us this uniformity was only an empirical law [see Chapter 16 here], i.e. if we didn’t know that the sun’s light and the earth’s rotation are the causes on which the periodical occurrence of daylight depends. We could still extend this empirical law to cases adjacent in time, but not across such a large distance of time as we can now ·with our knowledge of what the causes of the uniformity are·. Having evidence that the effects had been unaltered and precisely conjoined for five thousand years, we could infer that the unknown causes the conjunction depends on had existed—neither diminished nor counteracted—during that same period. So the same conclusions would follow as in the empirical-law case, except that in the latter we would only know that during five thousand years nothing had occurred to defeat perceptibly this particular effect; whereas when we know the causes (·i.e. in the real-law case·) we have the additional assurance that during that interval no such change has been noticeable in the causes themselves that could, if multiplied and continued long enough, defeat the effect.

Our knowledge of the causes enables us to judge whether any known cause could counteract them; whereas if we didn’t know them we couldn’t be sure that there weren’t causes actually in existence that could destroy them ·and thus break the day-night uniformity·. A bed-ridden savage who had never seen the Niagara Falls but who lived within hearing of them might imagine that the sound he heard would last forever; but if he knew it to be the effect of a rush of waters over a barrier of rock that is steadily wearing away, he would know that within a certain number of ages it will stop. Thus, the less we know about the causes on which the empirical law depends, the less sure we can be that it will continue to hold good; and the further we look into the future the more likely it is that some one of the causes that jointly give rise to the derivative uniformity will be destroyed or counteracted. The longer the time, the more chances there are of such an event—i.e. its not having occurred so far becomes less of a guarantee that it won’t occur within the given time. If, then, it is only to cases that are temporally adjacent (or nearly so) to the ones we have actually observed that any derivative law (not a law of causation) can be extended with an assurance equivalent to certainty, this is even more true of a merely empirical law. Fortunately, for our practical purposes we hardly ever have occasion to extend them further than that.

In respect of place, it might seem that a merely empirical law couldn’t be extended even to adjacent cases—i.e. that we couldn’t be sure of its being true in any place where it hasn’t been specially observed. The past duration of a cause guarantees its future existence unless something occurs to destroy it; but the existence of a cause in one place (or any number of places) doesn’t guarantee its existence in any other place. because there’s no uniformity in the collocations of primeval causes. Thus, when an empirical law is extended beyond the spatial limits within which it has been found true by observation, the cases to which it is being extended must be ones that are presumed to be within the influence of the same individual agents. If we discover a new planet within the known bounds of the solar system. . . .we can conclude with great probability that it revolves on its axis. All the known planets do so; and this uniformity points to some common cause, some event earlier than the first recorded astronomical observations ; and if Laplace is right in thinking that what is involved here is not merely the same kind of cause but the same individual cause (such as an impulse given to all the bodies at once), that cause—having acted at the extreme edges of the solar system—is likely (unless defeated by some counteracting cause) to have acted at every intermediate point and probably also somewhat beyond the limits. Which makes it likely to have acted on the supposed newly-discovered planet.

So when effects that are always found conjoined can be traced with any probability to a single cause (not merely a single kind of cause), we can with the same probability extend the empirical law of their conjunction to all places within the extreme spatial boundaries within which the fact has been observed (though allowing for the possibility of counteracting causes in some part of the field). And we can do this even more confidently when the law is not merely empirical, i.e. when the phenomena that we find conjoined are effects of known causes from whose laws we can deduce the conjunction of their effects. In that case, ·we have two advantages·. (i) We can extend the derivative uniformity over a larger space, because we can go beyond boundaries of our observation of the fact itself and include the extreme boundaries of the known influence of its causes. We know that the succession of day and night holds true of all the bodies in the solar system except the sun itself; but we know this only because we know what the causes of the day-night succession are. If we didn’t, we couldn’t extend the proposition beyond the orbits of the earth and moon. . . . (ii) We needn’t make as much allowance for the chance of counteracting causes. I have shown that our •loss of confidence because of the probability of counteracting causes should be proportional to our •ignorance of the causes on which the phenomena depend. . . .

Chapter 20. Analogy

§1. The word ‘analogy’, as the name of a mode of reasoning, is generally taken to name some kind of argument of an inductive nature but not amounting to a complete induction. But no word is used more loosely, or in a greater variety of senses, than ‘analogy’. It sometimes stands for arguments that could be presented as examples of the most rigorous induction. Whately, for instance. . . ., defines ‘analogy’ in a way that fits the meaning that mathematicians originally gave it, namely: resemblance of relations. In this sense, when a country that has sent out colonies is termed the ‘mother country’, the expression is analogical, signifying that the colonies of a country relate to it in the way children relate to their parents. And if any inference is based on this resemblance of relations—e.g. that obedience or affection is due from colonies to the mother country—this is called reasoning by analogy. And if it is argued •that a nation is best governed by an assembly elected by the people, from the admitted premise •that other associations for a common purpose, such as joint-stock companies, are best managed by a committee chosen by the relevant parties, this is again an argument from analogy in the sense I am examining. The premise is not

•that a nation is like a joint-stock company, or

•that Parliament is like a board of directors, but that

•Parliament relates to the nation in the way a board of directors relates to a joint-stock company.

. . . .Like other arguments from resemblance, an argument by analogy may •amount to nothing or •be a perfect and conclusive induction. The respect in which the two cases are alike may be the material one—the source of all the consequences that matter in the particular discussion. In the example last given, the resemblance is one of relation; the basis of the relation is the management by a few persons of affairs in which they and others have an interest. Someone may contend that this feature that is common to the two cases, along with the various consequences that follow from it, have the main share in determining all the effects that make up what we regard as good or bad administration. If he can establish this, his argument has the force of a rigorous induction; if he can’t, he is said to have ‘failed in proving the analogy’ between the two cases—a turn of phrase implying that when the analogy is proved the argument based on it can’t be resisted.

§2. But ‘analogical evidence’ is usually taken to cover any sort of resemblance (provided it doesn’t amount to a complete induction), without highlighting resemblance of relations. Analogical reasoning, in this sense, comes down to this:

•Two things resemble each other in one or more respects.

•A certain proposition is true of one of them, Therefore

•it is true of the other.

But that schema fits all reasoning from experience; nothing in it picks out analogy ·in particular· as distinct from induction ·in general·. In the strictest induction, equally with the faintest analogy, we argue that because A resembles B in one or more properties P1, it also resembles it in a certain other property P2. The difference is that in a complete induction it has been previously shown. . . .that there’s an invariable conjunction between P1 and P2, whereas in so-called analogical reasoning no such conjunction has been claimed. There has been no opportunity to use the Method of Difference or even the Method of Agreement; we merely conclude (and this is all that the argument of analogy amounts to) that a fact m that is known to be true of A is more likely to be true of B •if B agrees with A in some of its properties (even though no connection is known to exist between m and those properties), than •if no resemblance at all could be found between B and anything else known to possess the attribute m. [The switch from ‘fact’ to ‘attribute’ is Mill’s.]

This argument of course requires that the properties common to A and B are merely not known to be connected with m; they must not be properties known not to be connected with it. If we can. . . .show somehow that they have nothing to do with m, the argument of analogy is put out of court. The supposition the argument relies on is that m does depend on some property of A but we don’t know which. . . . After setting aside all the properties of A that we know to have nothing to do with m, there remain several that we can’t decide between; and B has one or more of these. We regard this as providing more or less strong grounds for concluding by analogy that B has the attribute m.

There can be no doubt that every such resemblance that can be pointed out between B and A provides some degree of probability, beyond what there would otherwise be, in favour of the conclusion drawn from it. If B resembled A in all its ultimate properties, its possessing the attribute m would be a certainty, not a probability; and every resemblance that can be shown to exist between A and B places the conclusion that much nearer to that point, i.e. to certainty. If A resembles B in having some ultimate property, there will be a resemblance between them in all the derivative properties flowing from that ultimate property, and m may be one of these. If A and B are alike •in some derivative property, there’s reason to expect that they are also alike •in the ultimate property from which that one derives, and ·therefore· •in the other derivative properties that depend on that same ultimate property. Every resemblance that can be shown to exist provides ground for expecting indefinitely many other resemblances; so the particular resemblance we are looking for will be found more often among things known to be alike than among things between which we know of no resemblance.

I might infer that there are probably inhabitants in the moon, because there are inhabitants on the earth, in the sea, and in the air; and this is the evidence of analogy. The property of having inhabitants is here assumed to be not •ultimate but •a consequence of other properties;. . . .but we don’t know which properties they are. Now, the moon resembles the earth in

•being a solid, opaque, nearly spherical substance,

•appearing to contain or to have contained active volcanoes;

•receiving about as much heat and light from the sun as our earth does;

•revolving on an axis;

•being composed of materials that gravitate, and obeying all the laws resulting from that property.

If this were all that was known regarding the moon, these various resemblances would make the thesis The moon has inhabitants more probable than it would otherwise be, though it would be useless to try to estimate how much more.

Along with the fact that

every resemblance proved between B and A in any respect that isn’t known to be irrelevant to the attribute m adds to the case for presuming that B has m,

it is clearly also true that

every dissimilarity proved between B and A ·in any respect that isn’t known to be irrelevant to the attribute m· creates a counter-probability of the same sort on the other side.

[Mill seems to mean by this ‘. . . adds to the case for presuming that B doesn’t have m’ rather than merely ‘. . . detracts from the case for presuming that B has m’.] It sometimes happens that different ultimate properties produce the same derivative property, but on the whole it is certain that things that differ in their ultimate properties will differ at least as much in the aggregate of their derivative properties, and that the unknown differences will bear some proportion to those that are known. So we will weigh •the known respects of alikeness between A and B against •their known respects of difference; and the answer to

‘Do the analogies between A and B count for or against B’s having the property m?’

will depend on which way the balance tilts. The moon is like the earth in the respects I have mentioned; but differs in

•being smaller,

•having a surface that is more uneven, and apparently volcanic throughout,

•having, at least on the side facing the earth, no atmosphere sufficient to refract light,

•having no clouds, and (it is therefore concluded) no water.

These differences, considered merely in themselves, might balance the resemblances, so that analogy wouldn’t provide any presumption either way. But some of the features that the moon lacks are, on the earth, indispensable conditions of animal life; so we can conclude that if life does exist in the moon (or at all events on the nearer side), its causes must be totally different from those on which animal life depends here—a consequence of the moon’s •differences from the earth not of its •similarities. Viewed in this light, all the resemblances between the moon and the earth count against the moon’s being inhabited. Because life can’t exist there in the way it does here, the greater the resemblance of the lunar world to the terrestrial one in other respects, the less reason we have to believe that the moon can contain life.

But the earth has a much closer resemblance to certain planets in our solar system—planets that have an atmosphere, clouds, consequently water (or some fluid analogous to it), and even give strong indications of snow in their polar regions; while temperature, though differing greatly on the average from ours, is in some parts of those planets, possibly not more extreme than in some habitable regions of our own. To balance these agreements, the known differences are chiefly in

•the average light and heat,

•speed of rotation,

•density of material,

•intensity of gravity,

and similar features of a secondary kind. With regard to these planets, therefore, the argument by analogy decidedly comes down in favour of their resembling the earth in its derivative properties such as that of having inhabitants; though when we consider how countlessly many their unknown properties are compared with the few that we know, we can’t attach any significant weight to any considerations of resemblance in which the known elements amount to so little compared with the unknown ones.

As well as competition between analogy and diversity, there can be a competition between conflicting analogies. The new case may be similar in some respects to cases in which m exists, and in other respects to cases in which it is known not to exist. Amber has some properties in common with vegetable products, others with mineral products. A painting of unknown origin may in some ways resemble known works of Titian while in others as strikingly resemble those of Raphael. A vase may bear some analogy to works of Grecian art and some to those of Etruscan or Egyptian. . . .

§3. So the value of an analogical argument inferring one resemblance from other resemblances without prior evidence of a connection between them depends on the extent of

(i) known resemblance

compared first with the extent of

(ii) known difference

and next with the extent of

(iii) the unexplored region of properties that might go either way.

It follows that where (i) the resemblance is very great, (ii) the known difference very small, and (iii) our knowledge of the subject-matter fairly extensive, the argument from analogy may be nearly as strong as the conclusion of a valid induction. If after much observation of B we find that it agrees with A in nine out of ten of its known properties, we can conclude with a probability of 9:1 that it will have any given derivative property of A. If we discover an unknown animal or plant closely resembling some known one in most of the properties we observe in it but differing in a few, we can reasonably expect to find in the unobserved remainder of its properties a general agreement with those of the former, but also a difference whose size corresponds proportionately to the amount of observed diversity.

We learn from this that the conclusions derived from analogy aren’t of much value unless the case toward which we are reasoning is adjacent—not near in place or time, but near in circumstances. In the case of effects whose causes are known imperfectly if at all, so that the observed order of their occurrence amounts only to an empirical law, it often happens that the conditions that have coexisted whenever the effect was observed have been very numerous. If a new case turns up in which these conditions don’t all exist though by far greater part of them do, with only a few lacking, the inference that the effect will occur—despite this absence of complete resemblance to the cases where it has been observed—may be highly probable, although this is only an argument from analogy. Of course no competent inquirer into nature will rest satisfied with this, however high its probability is, if a complete induction can be had, but will consider the analogy as a mere guide-post indicating the direction in which more rigorous investigations should be carried out.

It’s as guideposts that considerations of analogy have the highest scientific value. As I have remarked, analogical evidence doesn’t itself support any very high degree of probability except when the resemblance is very close and extensive; but any analogy, however faint, can be of the utmost value in suggesting experiments or observations that may lead to more positive conclusions. When the agents and their effects are out of the reach of further observation and experiment—as in the speculations about the moon and planets—such slight probabilities aren’t important, presenting merely an interesting theme for the pleasant exercise of imagination; but any suspicion that sets an able person to work devising an experiment, or providing a reason for trying one experiment rather than another, may be of the greatest benefit to science.

For this reason although I can’t accept as secure truths any of the scientific hypotheses that can’t be eventually brought to the test of actual induction—e.g. the two theories of light, the emission-·of-particles· theory of the last century and the undulatory theory that currently predominates—I can’t agree with those who regard such hypotheses as negligible. As is well said by David Hartley in his Observations on Man,. . . . ‘any hypothesis that has enough plausibility to explain a considerable number of facts helps us to digest these facts in proper order, to bring new ones to light, and make decisive experiments for the sake of future inquirers’. If an hypothesis •explains known facts and •has led to the prediction of others that were previously unknown and have since been verified by experience, the laws of the phenomenon x that is the subject of inquiry must be very like the laws of the class of phenomena to which the hypothesis assimilates x; and since an analogy that extends that far may well extend further, nothing is more likely to suggest experiments tending to throw light on the real properties of x than following out such an hypothesis. And this doesn’t require that the hypothesis be mistaken for a scientific truth. On the contrary, that illusion blocks the progress of real knowledge by leading inquirers to restrict themselves to the particular hypothesis that is most in favour at the time, instead of •looking out for every class of phenomena whose laws are in any way like the laws of x, and •trying all such experiments as might tend to the discovery of further analogies pointing in the same direction.

Chapter 21. Evidence for the law of universal causation

§1. I have now completed my review of the logical processes by which the laws or uniformities in the •sequence of phenomena, and uniformities in their •coexistence that depend on their laws of sequence, are ascertained or tested. As I recognised at the outset, and have shown more clearly as the investigation progressed, all these logical operations are based on the law of causation.

The validity of all the inductive methods depends on the assumption that every event—i.e. the beginning of every phenomenon—must have some •cause, some •antecedent whose existence it invariably and unconditionally follows. In the Method of Agreement this is obvious; that method openly proceeds on the supposition that we have found the true cause as soon as we have ruled out every other. Similarly with the Method of Difference. That method authorises us to infer a general law from two particular instances:

•one in which A exists together with many other circumstances, and B follows; and

•one in which A is absent while all the other circumstances remain the same, and B is prevented.

What does this prove? It proves that B in the particular instance can’t have had any cause other than A; but to infer from this that A was the cause, or that A will on other occasions be followed by B, is legitimate only the assumption that B must have had some cause, I.e. that among its antecedents in any single instance in which it occurs there must be one that has the capacity to produce it at other times. . . . There’s no need for me to spend time proving that this holds for the other inductive methods as well. The universality of the law of causation is assumed—not proved—in all of them.

Is this assumption justified? You may want to object:

No doubt most phenomena are connected as effects with some antecedent or cause, i.e. are never produced unless some assignable fact has preceded them; but the very fact that complicated inductive processes are sometimes needed shows that in some this regular order of succession isn’t apparent to our unaided apprehension [= ‘to our naked senses’]. So if the processes that bring these cases into the same category as the rest require us to assume the universality of the very law that they don’t at first sight appear to exemplify, isn’t this a petitio principii [see Glossary]? Can we prove a proposition by an argument that takes it for granted? And if it isn’t proved in that way, what is the evidence for it?

For this difficulty—which I have deliberately stated as strongly as possible—the school of metaphysicians who have long predominated in this country find a ready response. They say that the universality of causation is a truth that we can’t help believing—the belief in it is an instinct, one of the laws of our believing faculty. As the proof of this they say (and it’s all they have to say) that everyone does believe it; and they include it in their rather large catalogue of propositions that •can be logically argued against, and perhaps •can’t be logically proved, but •are of higher authority than logic, and •are so deeply built into the human mind that even someone who denies them in theory shows by his habitual practice that his arguments make no impression upon himself.

. . . .I protest against offering the disposition—however strong and however general—of the human mind to believe that P as evidence that P is true in external nature. Belief is not proof, and doesn’t dispense with the need for proof. I’m aware that to ask for evidence for a proposition that we’re supposed to believe instinctively is to expose oneself to the charge of ‘rejecting the authority of the human faculties’; and of course no-one can consistently do that. Why not? Because the human faculties are all that anyone has to judge by; and given that the meaning of the word ‘evidence’ is supposed to be ‘something that when laid before the mind induces it to believe’, to demand evidence when the belief is ensured by the mind’s own laws is appealing to the intellect against the intellect. That’s what they say. But I think this is a misunderstanding of the nature of evidence. What we mean by ‘evidence’ is not ‘anything and everything that produces belief’! Many things generate belief besides evidence. A mere strong association of ideas often causes a belief so intense as to be unshakable by experience or argument. Evidence isn’t what the mind does or must yield to, but what it ought to yield to because that will keep the mind’s belief conformable to fact. There is no appeal ·to any higher court· from the human faculties generally, but there is an appeal from one human faculty to another; from the judging faculty to those that attend to facts—i.e. the faculties of sense and consciousness. The legitimacy of this appeal is admitted whenever it is allowed that our judgments ought to fit the facts. To say that belief suffices for its own justification is making opinion the test of opinion; it is denying the existence of any outward standard that an opinion has to meet to count as true. We call one way of forming opinions ‘right’ and another ‘wrong’ because one does and the other doesn’t tend to make the opinion agree with the facts—to make people believe what really is, and expect what really will be. A mere disposition to believe that P, even if is instinctive, is no guarantee that P is true. If indeed the belief ever did amount to an irresistible necessity, there would be no use calling it into question because there would be no possibility of altering it. But even then it wouldn’t follow that the belief was true; it would only follow that mankind were under a permanent necessity of believing something that might be false. . . . But in fact there is no such permanent necessity. There is no proposition of which it can be said that every human mind must eternally and irrevocably believe it. Many of the propositions of which this is most confidently stated have in fact been disbelieved by many people. There are countless things of which it has been supposed that nobody could possibly help believing them, but no two generations would have the same list of them! One age or nation unquestioningly believes what to another seems incredible and inconceivable; one individual has no vestige of a belief that someone else thinks is absolutely built into human nature. None of these supposed instinctive beliefs is really inevitable. Everyone has the powers to develop habits of thought that make him independent of them; especially the habit of philosophical analysis, which is the best way to enable the mind to •command the laws of the merely passive part of its own nature, rather than •being commanded by them. This habit of thought shows us that things aren’t necessarily connected in fact because their ideas are connected in our minds, and is thus is able to loosen countless associations that reign despotically over the undisciplined or early-prejudiced mind—including associations that the school of thought I’m discussing thinks are born with us and instinctive. I’m convinced that anyone who •is accustomed to abstraction and analysis, •is willing to exert his faculties for this purpose, and •frees his imagination to make room for unfamiliar notions, will have no difficulty in conceiving that in some part. . . .of the universe events succeed one another at random without any fixed law. Nothing in our experience or in the nature of our minds constitutes any reason for believing that this is nowhere the case.

If the present order of the universe were brought to an end (which we’re perfectly able to imagine happening), starting off a chaos with •no fixed succession of events and •no clues to the future in the past, then if a human being miraculously survived to witness this change, he would surely stop believing in uniformity because there wouldn’t be any uniformity. If this is right, then the belief in uniformity either isn’t an instinct, or is one that can—like all other instincts—be conquered by acquired knowledge.

But there’s no need to speculate on what •might be, when we have certain knowledge of what •has been. It’s simply not true that mankind have always believed that all the successions of events were uniform and according to fixed laws. The Greek philosophers, even including Aristotle, recognised Chance and Spontaneity as among the agents in nature, so that for them there was no guarantee that the past had been similar to itself [presumably meaning ‘that each part of the past had resembled all its other parts’], or that the future would resemble the past. Even now at least half of the philosophical world, including the metaphysicians who most strenuously maintain that the belief in uniformity is instinctive, regard volitions as an exception to the uniformity and not governed by a fixed law.

·START OF A LONG FOOTNOTE·

Baden Powell’s Essay on the Inductive Philosophy contains an excellent passage which I’m glad to be able to quote, in confirmation of both the history and the doctrine that I have presented. Speaking of the ‘conviction of the universal and permanent uniformity of nature’, Powell writes: ‘This idea isn’t widely accepted and doesn’t grow in us naturally. Everyone on the basis of his experience comes to embrace a certain view of this kind—but it is limited to the thesis that what is going on around him at present, in his own narrow sphere of observation, will go on in the same way in future. The peasant believes that the sun that rose today will rise again tomorrow; that the seed put into the ground will be followed by the harvest this year as it was last year, and so on; but he has no notion of inferences like that on topics beyond his immediate observation. . . . And it’s not only the most ignorant who limit the truth in this way. There’s a general propensity to •believe that apart from specially ascertained laws of nature everything beyond common experience is left at the mercy of chance or fate or arbitrary intervention; and even to object to any attempted explanation by physical causes of an apparently unaccountable phenomenon.

‘So we have this ·limited· idea of the uniformity of nature; but how are we to generalise it? That task isn’t obvious, natural, or intuitive—far from it! It is utterly beyond the reach of most people. The fully universal notion of the uniformity of nature is a mark of the philosopher: it’s clearly the result of philosophical cultivation and training, and absolutely not the spontaneous offspring of any primary principle [see Glossary] naturally inherent in the mind, as some seem to believe. It is not a mere vague opinion taken up without examination as a common assumption to which we are always accustomed; on the contrary, all common prejudices and associations are against it. It is pre-eminently an acquired idea. It is not attained without deep study and reflection. The best informed philosopher is the man who most firmly believes it, even in opposition to received notions;

its acceptance depends on the breadth and depth of his inductive studies.’

·END OF FOOTNOTE·

§2. As I remarked here, our belief in the universality throughout nature of the law of cause and effect is itself an instance of induction; and by no means one of the earliest that any of us—let alone mankind in general—can have made. We arrive at this universal law by generalisation from many less general laws. We would never have had the notion of causation (in the philosophical meaning of the word) as a condition of all phenomena unless many cases of causation— i.e. many partial uniformities of sequence—had previously become familiar. The more obvious of the particular uniformities suggest and give evidence for the general uniformity, and once the general uniformity is established it enables us to prove the remainder of the particular uniformities of which it is made up. As, however, all rigorous processes of induction presuppose the general uniformity, our knowledge of the particular uniformities from which it was first inferred was, of course, derived not from •rigorous induction but from •the loose and uncertain procedure of inductio per enumerationem simplicem [see Glossary]; and because the law of universal causation is based on results obtained in that way it can’t itself rest on any better foundation. [Throughout that paragraph, ‘particular’ should have been ‘less general’.]

So it seems that induction per enumerationem simplicem, far from being an illicit logical process, is actually the only kind of induction possible; because the more elaborate process ·of sophisticated kinds of induction· depends for its validity on a law that is itself obtained in that inartificial way. [By ‘inartificial’ Mill means that induction per enumerationem simplicem doesn’t require skill, isn’t governed by complex rules, is (as its name indicates) simple.] Then isn’t there an inconsistency in contrasting the looseness of one method with the rigidity of another, when the rigid method is based on the looser one?

This inconsistency is only apparent. Of course if induction by simple enumeration were an invalid process, no process based on it could be valid—any more than we could rely on telescopes if we couldn’t trust our eyes. But it isn’t invalid; it’s merely fallible; and there are different degrees of fallibility. If we can substitute for the more fallible forms of a process an operation based on the same process in a less fallible form, that will be a very material improvement. And that’s what scientific induction does.

A procedure for drawing conclusions from experience must be regarded as untrustworthy when subsequent experience refuses to confirm it. By this criterion, induction by simple enumeration—i.e. generalisation of an observed fact from the mere absence of any known instance to the contrary—is in general a precarious and unsafe basis for confidence, because we’re constantly finding such generalisations to be false. Still, it provides some assurance. . . .for the ordinary guidance of conduct. It would be absurd to say that the generalisations arrived at by mankind at the outset of their experience—e.g. food nourishes, fire burns, water drowns—are not fit to be relied on.[1] There’s a scale of trustworthiness in the results of the original unscientific induction; and as I pointed out in chapter 4 the rules for the improvement of the process depend on the differences marked by this scale. The improvement consists in correcting one of these inartificial generalisations by means of another. This (I repeat myself here) is all that art can do. To test a generalisation by showing that it either follows from or conflicts with some stronger induction, some generalisation resting on a broader foundation of experience, is the beginning and end of the logic of induction.

§3. For any generalisation G reached by the method of simple enumeration, the broader (i.e. more general) G is, the less precarious it is. The process is misleading and inadequate exactly in proportion as the subject-matter of the observation is special and limited in extent. As the sphere widens, this unscientific method becomes less and less liable to mislead; and the most universal class of truths—including the law of causation and the principles of number and of geometry—are satisfactorily proved by that method and can’t be proved in any other way.

As applied to the uniformities that depend on causation, that remark follows obviously from the principles laid down in the preceding chapters. When a fact [see Glossary] has been observed several times to be true and never to be false, if we at once affirm it as a universal truth or law of nature—without testing it by any of the four methods of induction or deducing it from other known laws—we’ll usually err grossly; but we’re perfectly justified in affirming it as an empirical law, true within certain limits of time, place, and circumstance, provided the number of instances is greater than can plausibly be attributed to chance. Why not extend it beyond those limits? Because its holding true within them may be

an upshot of collocations that can’t be concluded to exist in one place because they exist in another;

or it may be

dependent on the accidental absence of counteracting agencies, which might be brought into play by any variation of time or the smallest change of circumstances.

With that in mind, now consider the case of a generalisation whose subject-matter is so widely diffused that every time, place, and combination of circumstances provides an example either of its truth or of its falsity, and suppose that it is never found to be otherwise than true. If the truth of this generalisation depends on collocations, they must be ones that exist at all times and places; and if it could be frustrated by any counteracting agencies, they must be ones that never actually occur! So it’s an empirical law that is coextensive [see Glossary] with all human experience; at which point the distinction between empirical laws and laws of nature vanishes, and the proposition takes its place among the most firmly established as well as broadest truths that science can discover.

Of all the generalisations that experience supports concerning the sequences and coexistences of phenomena, the most extensive in its subject-matter is the law of causation. It stands at the top of all observed uniformities—top in •universality, and therefore (if what I have been saying is right) top also in •certainty. . . . We’re justified in considering this fundamental law, though it was obtained by induction from particular laws of causation, as actually more certain than any of those from which it was drawn. It adds to them as much proof as it receives from them. Even the best established laws of causation are probably sometimes counteracted and thus suffer apparent exceptions; and this would have shaken mankind’s confidence in the universality of those laws if inductive processes based on the universal law hadn’t enabled us to attribute those exceptions to the agency of counteracting causes, thereby reconciling them with the law that they apparently conflict with. . . . When it comes to the ·universal· law of causation, on the other hand, we don’t know of any exceptions. And the exceptions that limit or apparently invalidate the special laws. . . .actually confirm the universal law: in all cases that are sufficiently open to our observation we can trace the difference of result— ·the apparent exception·—either to •the absence of a cause that had been present in ordinary cases, or to •the presence of one that had been absent; ·and this tracing involves the use of the universal law of causation·.

Because •the law of cause and effect is certain, it can pass its certainty on to all other inductive propositions that can be deduced from it; and the narrower inductions can be seen as getting their ultimate sanction from •that law, because every one of them x gains in certainty when we connect it with that larger induction and show that x can’t be denied, consistently with the law that everything that begins to exist has a cause. So we’re justified in the seeming inconsistency of

•holding induction by simple enumeration to be good for proving this general truth, which is the foundation of scientific induction, and yet

•refusing to rely on it for any of the narrower inductions.

I fully admit that if we didn’t know the law of causation we could still generalise the more obvious cases of uniformity in phenomena—always a bit precariously and sometimes extremely so—and this would give us a certain measure of probability. But there would be no need for us to estimate how probable such a result was, because ·we know in advance that· it never could amount to the degree of assurance that a proposition acquires when we show—by applying the four methods—that the supposition of its falsity is inconsistent with the law of causation. So we are •theoretically entitled and •practically required to disregard the probabilities derived from the early rough method of generalising, and not to consider a minor generalisation as proved unless the law of causation confirms it, or as probable unless we can reasonably expect it to be so confirmed.

§4. To assert both of these:

•Our inductive processes assume the law of causation.

•The law of causation is itself a case of induction.

is paradoxical only on the old theory of reasoning, according to which the universal truth (i.e. major premise) in a ratiocination is the real proof of the particular truths that are ostensibly inferred from it. According to the doctrine I presented in II.3.4 the major premise is not the proof of the conclusion; it is itself proved, along with the conclusion, from the same evidence. ‘All men are mortal’ is not the proof that Lord Palmerston is mortal; but our past experience of mortality authorises us to infer both the general truth and the particular fact, and both with exactly the same degree of assurance. Lord Palmerston’s mortality is inferred not •from the mortality of all men but •from the experience that proves the mortality of all men; and it’s a correct inference from experience if that general truth is correct. This relation between our general beliefs and their particular applications holds equally true in the more comprehensive case that we’re now discussing. Any new fact of causation inferred by induction is rightly inferred if it’s open to no objection that isn’t also an objection to the general truth that every event has a cause. The utmost certainty that can be given to a conclusion arrived at by inference stops at this point. When we have ascertained that the particular conclusion must stand or fall with the general uniformity of the laws of nature—is open to no doubt except doubt about whether every event has a cause—we have done all that can be done for it. The strongest assurance we can obtain of any theory respecting the cause of a given phenomenon is that the phenomenon has either that cause or none.

At a very early stage in our study of nature it might have been admissible to suppose ‘It has no cause’. But at the stage that mankind have now reached we can see that the generalisation that produces the law of universal causation has grown into a stronger and better induction, one deserving of greater reliance, than any of the subordinate generalisations. I think indeed that we can go a step further than this and regard the conclusion of that great induction as not merely •more certain than anything else but for all practical purposes •completely certain.

As I see it, there are two considerations that now give to the proof of the law of uniformity of succession this character of completeness and conclusiveness. (1) We now know it directly to be true of the great majority of phenomena; there are none of which we know it not to be true; the most that can be said about that is that there are some phenomena that we can’t—positively and from direct evidence—affirm it to be true of; but as phenomena become better known to us they are constantly passing from the ‘not known to be uncaused’ class into the ‘known not to be uncaused’ class; and for any phenomenon x whose cause we haven’t yet found, the absence of direct proof is accounted for by

•the rarity or the obscurity of x,

•our deficient means of observing x, or

•the logical difficulties arising from the complication of the circumstances in which x occurs;

so that even if x depends as rigidly on conditions as does any other phenomenon it wasn’t likely that we would know more about those conditions than we do. (2) There are phenomena the production and changes of which escape all our attempts to explain them in terms of any known law; but in every such case, the phenomenon or the objects involved in it are found in some instances to obey the known laws of nature. The wind, for example, is the type [see Glossary] of uncertainty and caprice, yet we sometimes find it obeying—with as much constancy as any phenomenon in nature—the law of the tendency of fluids to distribute themselves so as to equalise the pressure on every side of each of their particles; as in the case of the trade-winds and the monsoons. Lightning might once have been supposed to obey no laws; but now that we know it to be identical with electricity, we know that lightning in some of its manifestations is implicitly obedient to the action of fixed causes [see here]. I don’t think that there is any object or event in all our experience of nature—within the solar system, at least—that hasn’t either been •discovered by direct observation to follow laws of its own or •proved to be closely similar to objects and events which, in more familiar manifestations or on a more limited scale, follow strict laws. Our inability to trace the same laws on a larger scale and in the more specialised instances is explained by the number and complication of the modifying causes or by their inaccessibility to observation.

So the progress of experience has blown away the doubt there used to be about the universality of the law of causation, back when there were phenomena that seemed to be sui generis [see Glossary], not subject to the same laws as any other class of phenomena, and not yet found to have special laws of their own. But this great generalisation, back then, could reasonably have been—as in fact it was—acted on as highly probable before there were sufficient grounds for accepting it as a certainty. In this territory, as in everything, we don’t need and can’t have the absolute. We must hold even our strongest convictions with an opening left in our minds for the reception of facts that contradict them. Until we have taken this precaution, we aren’t entitled to act on our convictions with complete confidence just because no such contradiction appears. If something x has been found true in countless instances, and never found to be false after due examination, we are safe in acting on x as universally true, doing this provisionally until an undoubted exception appears; provided that this is a case where a real exception could scarcely have escaped our notice. When every phenomenon that we ever knew well enough to be able to answer the question had a cause on which it was invariably consequent, it was more rational to suppose that our inability to assign the causes of other phenomena arose from our ignorance than to think that some phenomena were uncaused, they being precisely the ones that we hadn’t yet had sufficient opportunity to study.

Notice, though, that the reasons for this reliance don’t hold in circumstances that we don’t know and can’t possibly have experience of. It would be folly to affirm confidently that this general law prevails in distant parts of the stellar regions, where the phenomena may be entirely unlike any we are acquainted with. . . . The uniformity in the succession of events, otherwise called ‘the law of causation’, must be accepted as a law not of the universe but only of the part of it that lies within the range of our means of sure observation, with some extension to adjacent cases. To extend it further is to make a supposition for which we have no evidence and can’t reasonably try to find any.

·LONG FOOTNOTE ENDING CHAPTER 21·

A rising thinker of the new generation in France, Hippolyte Taine, has given in the Revue des Deux Mondes the most masterly analysis, at least from one point of view, that has yet been made of this work ·of mine·. I mention him here because he assigns to the law of causation and to some other universal laws a certainty beyond the bounds of human experience, which I haven’t been able to accord to them.

He does this on the strength of our faculty of abstraction, which he seems to see as an independent source of evidence, not indeed as

•disclosing truths that aren’t contained in our experience, but as

•assuring us, as experience can’t, that truths we get from experience are universally true.

Taine seems to think that abstraction enables us not merely •to analyse the part of nature that we see, and exhibit separately the elements that pervade it, but also •to pick out those that are elements of the system of nature considered as a whole, not merely incidents belonging to our limited terrestrial experience. I’m not sure that I fully enter into Taine’s meaning; but I confess that I don’t see how any mere abstract conception extracted by our minds from our experience can be evidence of an objective fact in universal nature, evidence not provided by the experience itself. . . .

In an able article in the Dublin Review William George Ward contends that the uniformity of nature can’t be proved from experience, but only from ‘transcendental considerations’, and that physical science would have no basis if such transcendental proof were impossible.

When physical science is said to depend on the assumption that the course of nature is invariable, all that is meant is that the conclusions of physical science aren’t known as absolute truths: their truth is conditional on the uniformity of the course of nature; and all that the most conclusive observations and experiments can prove is that the result arrived at will be true if the present laws of nature are valid, and for as long as they are valid. But this is all the assurance we require for the guidance of our conduct. Ward doesn’t think that his transcendental proofs make our assurance practically greater, for he believes (as a Catholic) that the course of nature has been and frequently still is suspended by supernatural intervention.

All I needed to prove was this conditional conclusiveness of the evidence of experience, which is sufficient for the purposes of life; but I have ·gone further, and· given reasons for thinking that the uniformity ·of nature·, as itself a part of experience, is sufficiently proved to justify undoubting reliance on it. Ward challenges this, for three reasons.

(a) Even if it’s true that there has never yet been a well-authenticated case of a breach in the uniformity of nature, ‘the number of natural agencies constantly at work is incalculably large; and the observed cases of uniformity in their action must be immeasurably fewer than one-thousandth of the whole. Scientific men (let’s assume for the moment) have discovered that in a certain proportion of instances— immeasurably fewer than one-thousandth of the whole—a certain fact has prevailed, namely the fact of uniformity; and they haven’t found a single instance where that fact doesn’t prevail. Are they justified in inferring from these premises that the fact is universal? Surely the question answers itself!’ [Ward rubs this in with a ‘very grotesque’ example; we don’t need it to follow Mill’s reply.]

Ward’s argument doesn’t touch mine as it stands in the text. My argument is based on the fact

•that the uniformity of the course of nature as a whole is constituted by the uniform sequences of special effects from special natural agencies;

•that the number of these natural agencies in the part of the universe we know is not ‘incalculable’, not even extremely great;

•that we now have reason to think that most of them have been made sufficiently open to observation—if not separately, then at least in some of the combinations they enter into—for us to ascertain some of their fixed laws; and

•that this amount of experience justifies us in being as sure that the course of nature is uniform throughout as we used to be about the uniformity of sequence among the phenomena best known to us.

This view of the subject, if correct, destroys the force of Ward’s first argument.

(b) Next, he argues that many or most persons, both scientific and unscientific, believe that there are well authenticated cases of breaks in the uniformity of nature, namely miracles. This also fails to touch what I have said in the text. The only uniformity in the events of nature that I accept is the law of causation; and (as I shall explain in chapter 25) a miracle is not an exception to that law. In every case of an alleged miracle, a new antecedent is said to exist—a counteracting cause, namely the volition of a supernatural being. Thus, for anyone for whom beings with superhuman power over nature are a vera causa [see Glossary], a miracle is a case of the law of universal causation, not a deviation from it.

(c) Ward’s last argument (which he says is the strongest) is the familiar one of Reid, Stewart, and their followers—that whatever knowledge experience gives us of the past and present, it gives us none of the future. I confess that I can’t see any force in this argument. How does a future fact differ from a present or a past fact, except in their momentary relation to the human beings at present in existence? The answer made by Priestley in his Examination of Reid seems to me sufficient—namely that though we have had no experience of what is future, we have had abundant experience of what was future. The ‘leap in the dark’ (as Bain calls it) from the past to the future is exactly as much in the dark as the leap from a past that we have personally observed to a past that we haven’t. I agree with Bain’s opinion that the resemblance of what we haven’t experienced to what we have is by a law of our nature presumed through the mere energy of the idea, before experience has proved it. But this •psychological truth is not, as Ward seems to think in his criticism of Bain, inconsistent with the •logical truth that experience does prove it. The proof comes after the presumption, and consists in its invariable verification by experience when the experience arrives. . . .

In his Examination of Mr J. S. Mill’s Philosophy, James McCosh maintains that •the uniformity of the course of nature is a different thing from •the law of causation; and while he allows that the former is only proved by a long continuance of experience, and that it is not inconceivable or incredible that there may be worlds where it doesn’t prevail, he thinks that the law of causation is known intuitively. But the only uniformity in the events of nature is what arises from the law of causation; so as long as there remained any doubt that the course of nature was uniform throughout, at least when not modified by the intervention of a new (supernatural) cause, a doubt was necessarily implied not of the •reality of causation but of its •universality. If the uniformity of the course of nature has any exceptions—if any events succeed one another without fixed laws—to that extent the law of causation fails, and there are events that don’t depend on causes.

·END OF FOOTNOTE·

NOTES

[1] These early generalisations didn’t presuppose causation as scientific inductions do. What they did presuppose was uniformity in physical facts. But the observers were as ready to presume uniformity in the •coexistence of facts as in the •sequences of facts. On the other hand, they never thought of assuming that this uniformity was a principle pervading all nature: their generalisations didn’t imply that there was uniformity in everything, but only that as much uniformity as existed within their observation existed also beyond it. The induction fire burns doesn’t require for its validity that all nature should observe uniform laws, but only that there should be uniformity in. . . .the effects of fire on the senses and on combustible substances. And uniformity to this extent was not assumed, anterior to the experience, but proved by the experience. The same observed instances that proved the narrower truth proved the corresponding wider one. It’s because people lost sight of this fact, and thought that the law of causation in its full extent is necessarily presupposed in the very earliest generalisations, that they have been led to believe that the law of causation is known a priori and is not itself a conclusion from experience.

Chapter 22. Uniformities of coexistence that don’t depend on causation

§1. Phenomena occur either successively or simultaneously; so the uniformities in their occurrence are either uniformities of succession or uniformities of coexistence. Uniformities of succession are all covered by the law of causation and its consequences. Every phenomenon has a cause, which it invariably follows; and this gives rise to other invariable sequences among the successive stages of the same effect, as well as between the effects resulting from causes that invariably succeed one another.

In the same way a great variety of uniformities of coexistence also take their rise. Coordinate effects of a single cause naturally coexist with one another. High water at any point on the earth’s surface is uniformly simultaneous with high water at the diametrically opposite point, resulting from the directions in which the combined attractions of the sun and moon act on the oceans. An eclipse of the sun to us is invariably coexistent with an eclipse of the earth to a spectator on the moon, and their coexistence can also be deduced from the laws of their production.

So the question naturally arises: Can all the uniformities of coexistence among phenomena be explained in this way? Well, between phenomena that are themselves effects any coexistences must depend on the causes of those phenomena. If they are effects—immediately or remotely—of a single cause, the only way they can coexist is by virtue of some laws or properties of that cause; if they are effects of different causes, they must coexist because their causes coexist; and any uniformity of coexistence among these effects proves that those particular causes, within the limits of our observation, have uniformly been coexistent.

§2. But one class of coexistences can’t depend on causation, namely coexistences between things’ ultimate properties— the properties that cause all phenomena without themselves being caused by any phenomenon. (If they are caused, it is not by any phenomenon but by the origin of all things.) Yet among these ultimate properties there are coexistences, and indeed uniformities of coexistence. General propositions can be formed saying that whenever certain properties are found certain others are found along with them. We perceive water (for example) and recognise it to be water by certain of its properties P1. Having recognised it, we can ascribe to it countless other properties P2; and we couldn’t do that unless it were a general truth—a law or uniformity in nature—that the P1 properties always have the P2 properties conjoined with them.

In I.7.4 I explained in some detail what is meant by the Kinds of objects—classes that differ from one another not by a limited and definite set of distinctions but by an indefinite and unknown set of them. I now add that every proposition asserting something about a Kind affirms a uniformity of coexistence. All we know about any Kind is its properties; so the Kind, to us, is the set of properties by which it is distinguished from every other Kind.[1] In affirming anything of a Kind, therefore, we’re affirming something to be uniformly coexistent with the properties by which the kind is recognised; and that’s all that the assertion means.

All the properties of Kinds, then, can be counted amongst the natural uniformities of coexistence. They aren’t all independent of causation—only some of them. Some are ultimate properties, others derivative; for some no cause can be assigned, but others clearly depend on causes. Pure oxygen gas is a Kind, and one of its most straightforward properties is its gaseous form; but this property has for its cause the presence of latent heat; and if that heat were taken away (as has been done from many gases in Faraday’s experiments), the oxygen would lose its gaseous form along with many other properties that depend on—i.e. are caused by—that property.

Now for chemical compounds, which can be seen as resulting from the juxtaposition of substances that are different in Kind from themselves: there’s good reason to presume that a compound’s specific properties are effects of some of the properties of the elements ·that make it up·, though little progress has been made in tracing any invariable relation between any compound’s properties and the properties of its elements. There’s even more reason to make such a presumption when the object itself is not a primeval agent—·i.e. not an uncaused cause·—but an effect that depends on a cause or causes for its very existence. (Organisms are examples of this.) Thus, the only Kinds with properties that can confidently be regarded as ultimate are the ones that chemists call ‘simple substances’ or ‘elementary natural agents’; and the ultimate properties of these are probably much more numerous than we recognise, because every successful resolution of the properties of a chemical compound into simpler laws ·governing its elements· leads to the recognition of properties in the elements distinct from any previously known:

•The resolution of the laws of the motions of heavenly bodies established •the previously unknown ultimate property of mutual attraction between all bodies;

•the ongoing resolution of the laws of crystallisation, of chemical composition, of electricity, of magnetism etc. points to •various polarities that are ultimately inherent in the particles bodies are composed of;

•the resolution into more general laws of the uniformities in the proportions in which substances combine with one another led to the discovery of •the comparative atomic weights of different kinds of bodies;

and so forth. So the situation is this: every resolution of a complex uniformity into simpler and more elementary laws has an apparent tendency to reduce the number of the ultimate properties, and really does remove many properties from the list; but the result of this simplifying process is to trace an ever greater variety of different effects back to a single cause, and the further we go in this direction the more properties we are forced to recognise in a single object; and the coexistences of those properties must accordingly be ranked among the ultimate generalities of nature.

§3. So there are only two kinds of propositions that assert uniformity of coexistence between properties. If the properties depend on causes, the proposition that says they are coexistent is a derivative law of coexistence between effects, and it has the status of an empirical law until it’s resolved into the laws of causation on which it depends. . . . If the properties don’t depend on causes—i.e. are ultimate properties—then if it’s true that they invariably coexist they must all be ultimate properties of a single Kind; and it’s only of these that the coexistences can be classified as a special sort of laws of nature.

When we say that all crows are black, we assert a uniformity of coexistence. We assert that the property of blackness invariably coexists with the properties that define the class crow in common language or in our chosen scientific classification. Now, supposing blackness to be an ultimate property of black objects—i.e. supposing that it isn’t a result of causation, isn’t connected with antecedent phenomena by any law—then if all crows are black this must be an ultimate property of the kind crow or of some kind that includes it. If on the contrary blackness is an effect depending on causes, the proposition ‘All crows are black’ is clearly an empirical law; and what I have already said about empirical laws applies here too. [Mill ran ‘All crows are black’ in harness with ‘All negroes have woolly hair’, making exactly the same points regarding each.]

We have seen that in the case of all compounds—i.e. of everything except nature’s elementary substances and primary powers—the presumption is that the properties do really depend on causes; and it’s never possible to be certain that they don’t. So we wouldn’t be safe in claiming for any generalisation about the coexistence of properties a degree of certainty that wouldn’t be justified if the properties turn out to be the result of causes. A generalisation about coexistence, i.e. about the properties of Kinds, may be an ultimate truth, but it may be merely a derivative one; and if the latter, it is one of those derivative laws that •aren’t laws of causation and •haven’t been resolved into the laws of causation on which they depend; so it can’t be more evident than an empirical law can.

§4. We have found that a system of rigorous scientific induction can be applied to the uniformities in the •succession of phenomena; but nothing like that can be applied to the ultimate uniformities of •coexistence. The basis for such a system is lacking: there’s no general axiom relating to the uniformities of coexistence in the way the law of causation relates to the uniformities of succession. The Methods of Induction that can be used to discover causes and effects are based on the principle that everything x that has a beginning must have a cause; that among the circumstances that actually existed at the time x began there is one combination on which x is unconditionally consequent, and on the repetition of which x would certainly start again. But in an inquiry whether some Kind (such as crow) universally possesses a certain property (such as blackness) there is no room for any assumption analogous to this. We have no previous certainty that the property must have something that constantly coexists with it, i.e. must have an invariable coexistent in the same way that an event must have an invariable antecedent. When we feel pain, we must be in some state under which, if exactly repeated, we would always feel pain. But when we’re conscious of blackness, it doesn’t follow that there is something else present of which blackness is a constant accompaniment. So there is no room for elimination, no method of Agreement or Difference or Concomitant Variations. . . . We can’t conclude that the blackness we see in crows must be an invariable property of crows merely because there’s nothing else present of which it can be an invariable property! So we inquire into the truth of a proposition like ‘All crows are black’ under the same disadvantage as if, in our inquiries into the cause of x, we had to allow for the possibility that x occurred without any cause.

Overlooking this grand distinction was, it seems to me, the central error in Bacon’s view of inductive philosophy. He thought that the principle of elimination—that great logical instrument that he had the immense merit of first bringing into general use—could be applied (in the same sense and in as unqualified a manner) to the investigation of the coexistences of phenomena as to their successions. He seems to have thought that just as every event has a cause or invariable antecedent, so also every property of an object has an invariable coexistent, which he called its ‘form’. And his chosen examples to illustrate his method were inquiries into such forms: taking up objects that agree in some one general property—hardness or softness, dryness or moistness, heat or cold—he asked what else they have in common. Such inquiries couldn’t lead anywhere. The objects seldom have any such circumstances in common. They usually agree in the one property inquired into, and in nothing else. A great proportion of the properties that seem to us to be the likeliest to be really ultimate, seem to be inherently properties of many different Kinds of things that don’t have anything else in common. As for properties that we can give some account of because they are effects of causes, they have generally nothing to do with the ultimate resemblances or diversities in the objects themselves, but depend on some outward circumstances under the influence of which any object can manifest those properties. Certainly the case with Bacon’s favourite subjects of scientific inquiry—heat and coldness—as well as with hardness and softness, solidity and fluidity, and many other conspicuous qualities.

In the absence of any universal law of coexistence like the universal law of causation that regulates sequence, we’re thrown back upon the unscientific induction of the ancients—induction by simple enumeration where there are no counter-examples [Mill gives that phrase in Latin]. The reason we have for believing that all crows are black is simply that we have seen and heard of many black crows and never one of any other colour. We now face two questions: How far can this evidence reach? How are we to measure its strength in any given case?

§5. It sometimes happens that a mere change in the wording of a question, without changing its meaning, is a long step toward finding the answer. Our present topic is a case of this, I think. Here are two formulations for exactly the same thing:

•the degree of certainty of a generalisation for which our only evidence is that no counter-examples to it has yet been observed;

•the degree of improbability that a counter-example, if there were one, could have remained unobserved until now.

The reason for believing that all crows are black is measured by the improbability that crows of any other colour should have existed to the present time without our being aware of it. Let us state the question in this second way and consider: What is implied in the supposition that there may be crows that aren’t black? Under what conditions are we justified in regarding this as incredible?

If there really are non-black crows, one of two things must be the case. (i) The blackness in all crows hitherto observed is (as it were) an accident, not connected with any distinction of Kind. (ii) Blackness is a property of Kind, and non-black crows are a new Kind that we have overlooked although they fit the general description by which we have always recognised crows. We might prove (i) to be true if we casually discovered a white crow among black ones, or if we found that black crows sometimes turn white. And (ii) would be shown to be the fact if in Australia or Central Africa a species or a race of white or gray crows were found to exist.

§6. Supposition (i) implies that the colour is an effect of causation. If blackness in the crows in which it has been observed isn’t a property of Kind—so that an object can have it or lack it without any difference in its other properties— then it isn’t an ultimate fact in the individuals themselves but certainly depends on a cause. There are many properties that vary from individual to individual of the same Kind. . . . Some flowers can be either white or red without differing in any other respect. But these properties aren’t ultimate; they depend on causes. So far as a thing’s properties belong to its own nature and don’t arise from some external cause, they are always the same in the same Kind. Take, for instance, all •simple substances and •elementary powers, which are the only things of which we’re certain that at least some of their properties are ultimate. Colour is generally regarded as the most variable of all properties; but we don’t find that sulphur is sometimes yellow and sometimes white, or that it varies in colour at all except to the extent that colour is an effect of some external cause—e.g. the sort of light thrown upon it, or the mechanical arrangement of the particles after fusion, etc. We don’t find that iron is sometimes fluid and sometimes solid at the same temperature; gold sometimes malleable and sometimes brittle; that hydrogen sometimes combines with oxygen and sometimes not; or the like. If from simple substances we pass to any of their definite compounds such as water, lime, or sulphuric acid, there’s the same constancy in their properties. When properties vary from individual to individual, the individuals are either

•miscellaneous aggregations such as atmospheric air or rock, composed of heterogeneous substances and not belonging to any real Kind, or

•organisms.

In organisms there is great variability: animals of the same species and race, human beings of the same age, sex, and nationality will be extremely unalike, e.g. in face and figure. But there’s reason to believe that none of their properties are ultimate—that all of them are derivative, produced by causation. Why? Because •an organism is regulated by an extremely complicated system of laws, so that it’s open to being influenced by more (and more various) causes than any other phenomenon; and •the organism itself had a beginning, and therefore a cause. The presumption ·of nonultimateness· is confirmed by the fact that the properties that vary from one individual ·organism· to another also generally vary at different times in the same individual; any such variation, like any other event, has a cause and thus implies that the properties are not independent of causation.

So if blackness is merely accidental in crows, and can vary while the Kind remains the same, its presence or absence is doubtless not an ultimate fact but the effect of some unknown cause. If that is so, the universality of the experience that all crows are black is sufficient proof of a common cause, and establishes ‘All crows are black’ as an empirical law. Because there are countless instances in the affirmative, and so far none in the negative, the causes the property depends on must exist everywhere within the limits of the observations that have been made; and the proposition can be accepted as universally true within those limits, and with the permissible degree of extension to adjacent cases.

§7. In the second place—·i.e. picking up on item (ii) in §5·—if the property, in the instances in which it has been observed, is not an effect of causation, it is a property of Kind; and in that case the generalisation ‘All crows are black’ can’t be set aside except by the discovery of a new Kind of crow. But it’s not very improbable that a hitherto-undiscovered Kind should turn out to exist in nature—it happens often. We have no basis for trying to limit the Kinds of things that exist in nature. The only unlikelihood is the discovery of a new Kind in a region that we previously had reason to think we had thoroughly explored; and even this improbability depends on the how conspicuously different the newly-discovered Kind is from all the others. We often detect in the most frequented situations new Kinds of minerals, plants, and even animals that we had overlooked or confused with known species. On this second ground, therefore, as well as on the first, the observed uniformity of coexistence can only hold good as an empirical law, within the limits of actual observation as accurate as the nature of the case requires. That’s why it is that (as I remarked in chapter 3.3) we so often give up generalisations of this class at the first challenge. If any credible witness said he had seen a white crow, under circumstances that made it credible that it should have escaped notice until then, we would believe him.

So we find that uniformities in the coexistence of phenomena—those we have reason to regard as ultimate, as well as those that arise from the laws of causes that we haven’t yet detected—are entitled only to be accepted as empirical laws; and aren’t to be presumed true except within the limits of time, place, and circumstance in which the observations were made, or in strictly adjacent cases.

§8. We saw in chapter 21.3) that when empirical laws reach a certain point of generality they become as certain as laws of nature—or rather, at that point there’s no longer any distinction between •empirical laws and •laws of nature. As empirical laws approach this point—i.e. as they become more general—they also become more certain, so that their universality can be more strongly relied on. Even with the uniformities treated of in this chapter we can never be certain that they aren’t results of causation; and if they are, the more general they are the greater is the space in which

•the necessary collocations occur and

•no causes exist that could counteract the unknown causes on which the empirical law depends.

To say that P is an invariable property of some very limited class of objects is to say that P invariably accompanies some numerous and complex group of distinguishing properties; and this, if causation is at all concerned in the matter, indicates a combination of many causes and therefore a great openness to counteracting causes; while the comparatively narrow range of the observations makes it impossible for us to predict how widely unknown counteracting causes may be distributed throughout nature. But when a generalisation has been found to hold good of a very large proportion of all things whatever, it is already proved that most of the causes in nature have no power over it; that few changes in the combination of causes can affect it; because the majority of possible combinations must have already existed in some of the instances in which it has been found true. So if an empirical law is a result of causation, the more general it is the more it can be depended on. And even if it’s not a result of causation but is an ultimate coexistence, the more general it is the greater the amount of experience it is derived from, so the greater is the probability that if exceptions had existed some of them would already have shown up.

For these reasons, much more evidence is needed to establish an exception to one of the more general empirical laws than to establish an exception to a more special one. We could easily believe that there might be a new Kind of crow, i.e. a new Kind of bird resembling a crow in the properties we have until now regarded as distinctive of crow. It would be much harder to convince us of the existence of a Kind of crow having properties at variance with any generally recognised universal property of birds; and even harder if the properties conflicted with any recognised universal property of animals. And that fits the way of judging that is approved by mankind’s common sense and general practice; how incredulous people are about ·alleged· novelties in nature depends on how general the experience is that these novelties would contradict.

§9. It is conceivable that the alleged properties might conflict with some recognised universal property of all matter. Their improbability would be at the highest but it still wouldn’t amount to incredibility. There are only two known properties common to all matter,. . . .namely resistance to movement and gravitation. As Bain expresses it, inertia and gravity are coexistent through all matter, and proportionate ·to one another· in their amount. Neither of these properties, as he truly says, implies the other; and just for that reason we always have to allow that a Kind may be discovered having one of the two properties without the other. The hypothetical ‘ether’, if it exists, may be such a Kind. Our senses can’t recognise either resistance or gravity in it; but if the reality of a resisting medium should eventually be proved (e.g. by alteration in the times of revolution of comets, combined with evidence provided by the phenomena of light and heat), it would be rash for us to conclude from this alone, without other proofs, that it must gravitate.

Even the greater generalisations, which concern comprehensive Kinds that include a great number and variety of lowest species, are only empirical laws that rest merely on induction by simple enumeration and not on any process of elimination—a process inapplicable to this sort of case. Such generalisations, therefore, ought to be based on an examination of all the lowest species covered by them—not just some of them. Just because a proposition is true of a number of animals we can’t conclude that it is therefore true of all animals. If anything P is true of two species x and z that differ more from one another than either differs from a third species y, especially if y occupies in most of its known properties a position between x and z, there’s some probability that P will also be true of y; for it is often (not always) found that there’s a sort of parallelism in the properties of different Kinds, and that their degree of unlikeness in one respect bears some proportion to their unlikeness in others. We see this parallelism in the properties of the different metals; in those of sulphur, phosphorus, and carbon; of chlorine, iodine, and bromine; in the natural orders of plants and animals, etc. But there are countless anomalies and exceptions to this sort of conformity—if indeed the conformity itself is anything but an anomaly and an exception in nature.

So we learn this about universal propositions that •concern the properties of superior Kinds and •are not based on proved or presumed causal connection: they ought not to be hazarded until one has separately examined every known sub-kind included in the larger Kind; and even then they must be held in readiness to be given up when some new anomaly turns up, which is likely enough to happen, even with the most general of these empirical laws. Thus, the many universal propositions that people have tried to lay down concerning •simple substances or •any of the classes that have been formed among simple substances have either faded into emptiness with the progress of experience or been proved to be erroneous; and each Kind of simple substance remains with its own collection of properties apart from the rest, apart from a certain parallelism with a few other Kinds that are the most similar to itself. In organisms, indeed, many propositions have been ascertained to be universally true of genera that are that are high in the classification table, and to many of these the discovery of exceptions is extremely improbable. But these, as I said already, we have every reason to believe that these properties depend on causation ·and therefore lie outside the scope of this paragraph·.

Uniformities of coexistence, then, not only when they follow from laws of succession but also when they are ultimate truths, must for logical purposes be classified among empirical laws, and fall under exactly the same rules as the unresolved uniformities that are known to depend on causation.

NOTES

[1] A Kind may be identified by some one remarkable property: but usually it takes several, each separate property being shared with other Kinds. The diamond’s colour and brightness are shared with the paste from which false diamonds are made; its octohedral form is shared with alum and magnetic iron ore; but the colour and brightness and form together identify its Kind—i.e. are a sign to us that it is combustible, that when burned it produces carbonic acid, that it can’t be cut with any known substance, along with many other ascertained properties and the fact that there are indefinitely many still unascertained.

Chapter 23. Approximate generalisations. Probable evidence

§1. As well as generalisations from experience that profess to be universally true there are inductive truths that don’t claim to be universal—don’t say that the predicate is always true of the subject—but which are nevertheless extremely valuable. An important part of the field of inductive knowledge consists not of universal truths but of approximations to such truths; and when a conclusion is said to rest on probable evidence, the premises it is drawn from are usually generalisations of this sort.

Just as every certain inference about a particular case implies that there is ground for a general proposition of the form ‘Every A is B’, so also every probable inference supposes that there’s ground for a proposition of the form ‘Most A are B’; and in an average case the degree of probability of the inference will depend on the proportion between •the number of instances existing in nature that accord with the generalisation and •the number that conflict with it.

§2. Propositions of the form ‘Most A are B’ are much less important in science than in everyday life. To the scientific inquirer they are valuable mainly as stepping-stones to universal truths. The discovery of universal truths is the proper end of science; its work isn’t done if it stops at the proposition that a majority of A are B, without providing some way of marking off that majority from the minority. As well as being •relatively imprecise and •impossible to apply confidently to individual cases, these imperfect generalisations are •almost useless as means of discovering ulterior truths through deduction. Admittedly we can infer ‘Most A are C’ from ‘Most A are B’ and ‘Every B is C’; but in most cases where a second proposition of the approximate kind is introduced—or even when there’s only one and it is the major premise—nothing can be positively concluded. When the major is ‘Most B are D’ then even if the minor is ‘Every A is B’ we can’t infer that most A are D; we can’t even infer with any certainty that some A are D. Though the majority of the class B have the attribute signified by D, the whole of the sub-class A may belong to the minority.[1]

For practical guidance, however, approximate generalisations are often all we have to rely on. Even when science has discovered the universal laws of a phenomenon, they don’t serve our everyday purposes. •They are usually too cluttered with conditions to be suitable for everyday use; and •the cases that turn up in ordinary life are too complicated, and our decisions have to be taken too rapidly, to allow us to wait until the existence of a phenomenon can be proved by what have been scientifically ascertained to be universal marks of it. To be indecisive and reluctant to act because we don’t have perfectly conclusive evidence to act on is a defect sometimes found in scientific minds, and when that happens it makes the mind in question unfit for practical emergencies. If we want to succeed in action, we must judge by indications that sometimes (though not usually) mislead us, and we must try to make up for the incomplete conclusiveness of one indication by obtaining others to corroborate it. So the principles of induction applicable to approximate generalisations are as important a subject of inquiry as the rules for the investigation of universal truths. You might reasonably expect the former inquiry to occupy nearly as much of this book as the latter, ·but in fact it won’t, because· the principles governing approximate generalisations are mere corollaries of the principles I have already discussed—namely the principles governing universal propositions.

§3. There are two sorts of cases where we have to steer by generalisations of the form ‘Most A are B’. (i) They are all we have; we haven’t been able to carry our investigation of the laws of the phenomena any further. For example:

•Most dark-eyed persons have dark hair;

•Most springs contain mineral substances;

•Most stratified formations contain fossils.

This class of generalisations isn’t very important, and here is why. It often happens that we see no reason why what’s true of most individuals in A isn’t also true of the remainder, and we can’t find a general description that marks off the ones of which it is true from the remainder, yet if we will settle for propositions that are less general and will break down the class A into sub-classes, we can generally obtain a collection of propositions that are exactly true. We don’t know why most wood is lighter than water, nor can we point out any general property marking off wood that is lighter than water from wood that is heavier. But we know exactly which species are the one and which the other. . . .

(ii) It often happens, however, that ‘Most A are B’ is not the peak of our scientific attainments, though the knowledge we have that goes further can’t conveniently be brought to bear upon the particular instance. Even when we know what circumstances distinguish the part of A that has B from the part that doesn’t, it can happen in an individual case that we don’t have the means (or don’t have time) to examine whether those characteristic circumstances exist or not. This is generally our situation when the inquiry is of the kind called ‘moral’, i.e. the kind that aims to predict human actions. If we are to affirm universally anything about the actions of classes of human beings, the classification must be based on the circumstances of their mental culture and habits, which in an individual case are seldom exactly known; and classes based on these distinctions would never exactly coincide with the classes into which mankind are divided for social purposes. All propositions about the actions of human beings as ordinarily classified, or as classified according to any kind of external indications, are merely approximate. We can only say ‘Most persons of a particular age, profession, country, or rank in society, have such-and-such qualities’; or ‘Most persons, when placed in certain circumstances, act in such-and-such a way’. We often know well enough what causes the qualities depend on, or what sort of persons they are who act in that particular way; but we seldom have the means of knowing whether any individual person has been under the influence of those causes, or is a person of that particular sort. We could replace the approximate generalisations by universally true propositions; but these would hardly ever be applicable in practice. We would be sure of our majors, but we wouldn’t be able to get minors to fit; so we are forced to draw our conclusions from coarser and more fallible indications.

§4. An approximate generalisation can be accepted only as an empirical law. Propositions of the form ‘Every A is B’ aren’t necessarily laws of causation, or ultimate uniformities of coexistence; propositions like ‘Most A are B’ necessarily aren’t so. Propositions that have been true in every observed instance needn’t follow necessarily from laws of causation; and if they don’t, they may for all we know be false beyond the limits of our observation; and this holds even more obviously for propositions that are true only in a mere majority of the observed instances.

How certain we can be of the proposition ‘Most A are B’ depends in part on whether (i) that approximate generalisation is the whole of our knowledge of the subject or (ii) it isn’t. In the case (i) we know only that most A are B, not why they are so nor in what respect those that are B differ from those that aren’t. Then how did we learn that most A are B? In exactly the way in which we would have learned that all A are B if that had been the fact of the matter. We collected enough instances to rule out chance, and then compared the number of affirmative instances with the number of negative ones. The result, like other unresolved derivative laws, can be relied on only within the limits of place and time and circumstance under which its truth has been observed. ·Why ‘and circumstance’?· Because we are ignorant of the causes that make the proposition true, so we can’t tell how any new circumstance might affect it. The proposition ‘Most judges can’t be swayed by bribes’ would probably be found true of Englishmen, Frenchmen, Germans, North Americans, and so forth; but if on this evidence we extended the assertion to Orientals we would be overstepping the limits, not only of place but of circumstance, within which the fact had been observed, and would let in possibilities of the absence of the determining causes or the presence of counteracting ones that might be fatal to the approximate generalisation.

(ii) When the approximate proposition is not the peak of our scientific knowledge but only the most available form of it for practical guidance—when we know not only that most A have the attribute B but also the causes of B or some properties that mark off the portion of A that has B from the portion that doesn’t—we are better placed than we were in (i). Now we have two ways of ascertaining whether it’s true that most A are B:

•the direct way, as in (i), and

•an indirect way, namely examining whether the proposition can be deduced from the known cause of B or from any known criterion of B.

Consider the question ‘Is it true that most Scotchmen can read?’ We and our informants may not have observed a sufficient number and variety of Scotchmen to ascertain this fact; but when we consider that the ability to read is caused by being taught to read, another way of answering the question presents itself, namely inquiring whether most Scotchmen have been sent to schools where reading is effectively taught. Sometimes one of these two approaches is the more available, sometimes the other. . . . It often happens that neither can yield as satisfactory an induction as could be desired, and that the grounds on which the conclusion is accepted are compounded of both. . . .

[Mill adds a paragraph saying that it is sometimes right for us to go beyond ‘Most A are B’ when we know enough to do so. Should we believe this witness to the crime? We wouldn’t want to answer that simply on the grounds that ‘Most persons on most occasions speak the truth’. He concludes the section:] It seems unnecessary to spend longer on the question of the evidence of approximate generalisations; so I’ll proceed to an equally important topic, that of the cautions to be observed in arguing from these incompletely universal propositions to particular cases.

§5. There’s no difficulty about this when it’s a matter of directly applying an approximate generalisation to an individual instance. If ‘Most A are B’ has been established, by a sufficient induction, as an empirical law, we can conclude that This particular A is B with a probability based on the preponderance of the number of affirmative instances over the number of exceptions. If we have numerical precision in the data, we can have equal precision about the chances of error in the conclusion. If we have established as an empirical law that nine out of every ten A are B there will be one chance in ten of error in assuming that any given A is a B; but this holds only within the same limits of time, place, and circumstance as bounded the observations, so it can’t be counted on for any sub-class or variety of A (or for A in any set of external circumstances) that weren’t included in the average. We can guide ourselves by the proposition Nine out of every ten A are B only in cases of which we know only they are within the class A. If we know that a particular instance i not only that it belongs to A but also what species or variety of A it belongs to, we’ll usually go wrong in applying to i the average we have found for the whole genus A, because the average corresponding to that species alone would probably differ from it materially. Similarly, if i, instead of being a particular sort of instance, is an instance known to be affected by a particular set of circumstances, it would again probably be misleading to apply to i the same probability of being B as holds on average for all of A’s members. A general average should be applied only to cases that aren’t known to be, and can’t be presumed to be, other than average cases. Such averages, therefore, are usually of little practical use except in affairs that concern large numbers. Tables of life-expectancy are useful to insurance offices, but they don’t go far towards informing you about your life-expectancy or me about mine, because almost everyone has a life-expectancy that is either better or worse than the average. Such averages merely supply the first term in a series of approximations, the subsequent terms reflecting growing knowledge of the circumstances of the particular case.

§6. From the application of a single approximate generalisation to individual cases, I proceed to the application of two or more of them together to the same case.

When a judgment J applied to an individual instance is based on the conjunction of two approximate generalisations P_1 and P_2, the latter may support J in two different ways. (a) In one, P_1 and P_2 are each separately applicable to the case in hand, and we combine them so as to give to J the double probability arising from P_1 and P_2 separately. This could be called joining two probabilities by way of addition; it gives to J a greater probability than either P_1 or P_2 has. (b) The other occurs when P_1 is directly applicable to the case, P_2 being applicable to it only by virtue of the application of P_1. This is joining two probabilities by way of ratiocination or deduction; it gives to J a lower probability than either P_1 or P_2 has. The type of (a) is

Most A are B;

Most C are B;

This thing is both an A and a C; therefore

This thing is probably a B.

The type of (b) is

Most A are B;

Most C are A;

This thing is a C; therefore

This thing is probably an A, therefore

This thing is probably a B.

Examples of (a): the guilt of the accused man is inferred from •the testimony of two unconnected witnesses, or from •the evidence of two incriminating facts—e.g. he concealed himself, and his clothes were stained with blood. Examples of (b): the man’s guilt is inferred from •one witness’s testimony about what he heard another person say, or from •the fact that he washed or destroyed his clothes, which is supposed to make it probable that they were stained with blood. Instead of only two links, as in these instances, there can be chains of any length. . . .

(a) When approximate generalisations are joined by way of addition, we can use the theory of probabilities laid down in chapter 17 to work out how each of them adds to the probability of a conclusion that has the support of them all.

If on average two of every three As are Bs, and three of every four Cs are Bs, what is the probability that something that is both an A and a C is also a B? [Mill presents his answer to this in two rather obscure plain-language versions, and then more clearly thus:] The chance that an A is not a B is 1/3, the chance that a C is not a B is 1/4; hence if the thing is both an A and a C, the chance of its not being a B is 1/3×1/4 = 1/12, and the chance of its being a B is 11/12.[2]

This computation assumes of course that the probabilities arising from A and C are independent of each other. There mustn’t be any connection between A and C such that a thing’s belonging to one affects the probability of its belonging to the other. Otherwise the not-Bs that are Cs may be, most or even all of them, identical with the not-Bs that are As; in which last case the probability arising from A and C together will be no greater than that arising from A alone.

(b) When approximate generalisations are joined together by way of deduction, the probability of the conclusion lessens at each step. From two premises such as Most A are B and Most B are C we can’t with certainty conclude that even a single A is C; for the whole of the portion of A that falls under B may be contained in the exceptional part of B, ·the part that doesn’t fall under C·. Still, those two propositions provide an appreciable probability that any given A is C, provided the average on which Most B are C is based wasn’t biased by any reference to Most A are B. That is, the proposition Most B are C must have been arrived at in a manner leaving no suspicion that the probability arising from it is not fairly distributed over the section of B that belongs to A. For though the instances that are A could be all in the minority, they also could be all in the majority; and these two possibilities cancel out. On the whole, the probability arising from the two propositions taken together will be correctly measured by the probability arising from the one multiplied by the probability arising from the other. If nine out of ten Swedes have light hair, and eight out of nine inhabitants of Stockholm are Swedes, the probability arising from these two propositions that any given inhabitant of Stockholm is light-haired will amount to 8/10, though it is possible that the whole Swedish population of Stockholm belongs to that tenth of the people of Sweden who don’t have light hair.

[Where this paragraph has ‘Let the proposition Most A are B be true because nine-tenths of the As are B’, what Mill actually wrote was ‘Let the proposition, Most A are B, be true of nine in ten’. That doesn’t make sense: there’s no way ‘Most A are B’ could be true of any individual A or, therefore, of nine As out of every ten. Other occurrences of this slip are silently corrected.] If the premises are known to be true not because of •a bare majority of their respective subjects but because of •nearly the whole, we can go on joining one such proposition to another for several steps before reaching a conclusion that isn’t presumably true even of a majority. The error of the conclusion will amount to the sum of the errors of all the premises. Let the proposition Most A are B be true because nine-tenths of the As are B, and let Most B are C be true because eight-ninths of Bs are C; then not only will one A in ten not be C, because not B, but even of the nine-tenths that are B only eight-ninths will be C; i.e. the cases of A that are C will be only 8/9×9/10 = 72/90 = 4/5. Let us now add Most C are D and suppose this to be true because seven-eighths of Cs are D; the proportion of A that is D will be only 7/8×8/9×9/10 = 7/10. Thus the probability progressively dwindles. But we usually can’t measure the lessening of probability that occurs at each step, because the experiences on which our approximate generalisations are based usually can’t be numerically estimated. So we have to settle for remembering •that it does diminish at every step, and •that the conclusion after a few steps is worth nothing unless the premises are extremely close to being universally true. A hearsay of a hearsay, or an argument from supposed evidence that depends not on immediate marks but on marks of marks is worthless at a very few removes from the first stage.

§7. There are, however, two cases in which reasonings depending on approximate generalisations can be carried to any length we please •with as much assurance as if they were composed of universal laws of nature and •with no departure from strictly scientific standards. . . . These are cases where the approximate generalisations are, for purposes of ratiocination, as suitable as if they were complete generalisations, because they can be transformed into exactly equivalent complete generalisations.

(i) If our reason for stopping at ‘Most As are B’ is not the impossibility but only the inconvenience of going further—i.e. if we know what marks off the As that are B from those that aren’t—we can replace the approximate proposition by a universal proposition with a proviso. The proposition

‘Most persons who have uncontrolled power employ it badly’

is a generalisation of this sort, and can be replaced by

All persons who have uncontrolled power employ it badly provided they don’t have unusual strength of judgment and rectitude of purpose.

The proposition carries the hypothesis or proviso with it, so it can be dealt with not as an approximate proposition but as a universal one. However many steps the reasoning takes, the proviso is carried along to the conclusion and indicates exactly how far the conclusion is from being applicable universally. If other approximate generalisations are introduced along the way, each of them also being expressed as a universal proposition with a proviso attached, the sum of all the provisos will appear at the end as the sum of all the errors that affect the conclusion. To the indented proposition a few lines back let us add

All absolute monarchs have uncontrolled power unless they need the active support of their subjects

(as was the case with Queen Elizabeth, Frederick of Prussia, and others). Combining these two propositions we can deduce a universal conclusion that will be subject to both the provisos in the premises:

All absolute monarchs employ their power badly unless they need the active support of their subjects, or unless they are persons of unusual strength of judgment and rectitude of purpose.

It doesn’t matter how rapidly the provisos in our premises accumulate, as long as we can in this way record each of them and keep an account of the aggregate as it swells up.

(ii) There is a case where approximate propositions count for scientific purposes as universal ones, even if we don’t know the conditions that mark off the ‘most’ from the others. This occurs when we are studying the properties not of •individuals but of •multitudes. The main one is the science of politics, or of human society. This science is principally concerned with the actions not of solitary individuals but of masses, with the fortunes not of single persons but of communities. For a statesman it is generally enough to know that most persons act or are acted on in a particular way; since his theorising and his practical arrangements refer almost exclusively to cases in which the whole community, or some large portion of it, is acted on all at once, so that what is done or felt by most persons determines what the body at large does or undergoes. He can get on well enough with approximate generalisations on human nature, since what is true approximately of all individuals is true absolutely of all masses. [That striking sentence is verbatim from Mill.] And even when the conduct of individual men have a part to play in the statesman’s deductions—e.g. when he is reasoning about kings or other single rulers—still he must in general both reason and act as if what is true of most persons were true of all, because he is providing for indefinite duration involving an indefinite succession of such individuals.

Those two considerations are a sufficient refutation of the popular error that theorising about society and government, because it rests on merely probable evidence, must be less certain and scientifically accurate than the conclusions of what are called the exact sciences, and less reliable in practice. There are reasons enough why the sciences dealing with human behaviour must remain inferior to at least the more perfect of the physical sciences; why the laws of their more complicated phenomena can’t be so completely deciphered, or their phenomena predicted with as much assurance. But though we can’t attain to so many truths, there is no reason why those we can attain should deserve less reliance, or have less of a scientific character. I’ll drop this topic now, and return to it in Book VI.

NOTES

[1] De Morgan in his Formal Logic rightly says that from ‘Most A are B’ and ‘Most A are C’ we can infer with certainty that some B are C. But this is the utmost limit of the conclusions that can be drawn from two approximate generalisations whose precise degree of approximation to universality is unknown or undefined.

[2] [Mill has here a long footnote in which he reports an objection that ‘a mathematical friend’ made to this paragraph. He states the reasoning behind the objection and admits that in the seventh edition of this work ‘I accepted this reasoning as conclusive. More attentive consideration, however, has convinced me that it contains a fallacy.’ He is right, and we needn’t go through all this. Here’s its last paragraph:] The true theory of the chances is best found by going back to the scientific grounds on which the proportions rest. The degree of frequency of a coincidence depends on, and is a measure of, the-frequency-combined-with-the effectiveness of the causes that are favourable to it. If out of every twelve As taken indiscriminately eight are Bs and four are not, this implies that

there are causes operating on each A that tend to make it a B, and these causes are sufficiently constant and powerful to succeed in eight out of twelve cases, but fail in the remaining four.

So if out of twelve Cs nine are Bs and three are not, it must be the case that

there are causes operating on each C that tend to make it a B, and these causes succeed in nine cases and fail in three.

Now suppose twelve items that are both As and Cs. The whole twelve are now operated on by both sets of causes. One set is sufficient to prevail in eight of the twelve cases, the other in nine. The analysis of the cases shows that six of the twelve will be Bs through the operation of both sets of causes; two more in virtue of the causes operating on A; and three more through those operating on C, and that there will be only one case in which all the causes will be inoperative. The total number, therefore, which are Bs will be eleven in twelve, and the evaluation in the text is correct.

Chapter 24. The remaining laws of nature

§1. I showed in I.5 that all the assertions that can be conveyed by language express one or more of five things:

existence

order in place

order in time

causation

resemblance.

Causation, on my view of it, isn’t fundamentally different from order in time, so the five species of possible assertions are reduced to four. The present Book up to here has been concerned with order in time in each of its two modes, coexistence and succession. And now I have finished with that topic insofar as it falls within the limits assigned to this work, discussing the nature of the evidence on which order-in-time propositions rest, and the processes of investigation by which they are ascertained and proved. There remain three classes of facts—existence, order in place, and resemblance—in regard to which the same questions are now to be answered.

Little needs to be said about existence in general, which is a topic not for logic but for metaphysics. To determine what things can be recognised as really existing independently of our own states of mind, what ‘exists’ means as applied to such things, belongs to the consideration of ‘Things in themselves’, a topic that I have kept at as great a distance as possible throughout this work. Existence, so far as logic is concerned about it, has reference only to phenomena—to actual or possible states of external or internal consciousness in ourselves or in others. The only things whose existence can be a subject of logical induction are the feelings of beings that have them, or the possibilities of having such feelings; because those are the only things whose existence in individual cases can be a subject of experience [= ‘can be known through experience’].

It’s true that we say a thing ‘exists’ even when it is absent and therefore can’t be perceived. But then its ‘existence’ is to us only another word for our conviction that we would perceive it. . . .if we were in the appropriate circumstances of time and place and had perfect sense-organs. My belief that the Emperor of China exists is simply my belief that if I were transported to the imperial palace or some other locality in Peking I would see him. My belief that Julius Caesar existed is my belief that I would have seen him if I had been present ·at an appropriate time· in the senate-house at Rome. When I believe that stars exist further away than I can see even with help from the most powerful telescopes yet invented, my belief, philosophically expressed, is that with still better telescopes I could see them, or that they could be perceived by beings closer to them in space or equipped with better eyesight than mine.

So a phenomenon’s ‘existence’ is simply another word for its being perceived or for the inferred possibility of its being perceived. When the phenomenon is within the range of present observation, that’s how we assure ourselves of its existence; when it is beyond that range and is therefore said to be ‘absent’, we infer its existence from marks or evidences. These evidences are other phenomena that are ascertained by induction to be connected—either in succession or in coexistence—with the given phenomenon. So the simple existence of an individual phenomenon, when it’s not directly perceived, is inferred from some inductive law of succession or coexistence; and consequently it can’t be brought under any inductive principles that are special to itself. We prove the existence of a thing by proving that it is connected by succession or coexistence with some known thing.

General propositions of this class, i.e. ones affirming the bare fact of existence, have a special feature that makes the logical treatment of them a very easy matter—namely, being generalisations that are sufficiently proved by a single instance. That ghosts or unicorns or sea-serpents exist would be fully established if it could be ascertained definitely that such things had been seen even once. Whatever has once happened can happen again; the only question relates to the conditions under which it happens.

With simple •existence, therefore, inductive logic has no knots to untie. So we can move on to the remaining two great classes into which facts have been divided. •resemblance and •order in place.

§2. Resemblance and its opposite are seldom regarded as subjects of science (except when they take the form of equality and inequality). They’re supposed to be perceived by simple apprehension; by merely applying our senses or directing our attention to the two objects at once, or in immediate succession. And this simultaneous (or virtually simultaneous) application of our faculties to the two things that are to be compared is indeed the ultimate appeal wherever it can be done; but in most cases the objects can’t be brought so closely together that a complete feeling of their resemblance directly arises in the mind. All we can do is to compare them with some third object that can be transported from one to the other. And even when the objects can be set side by side, we don’t have a perfect knowledge of their resemblance or difference unless we compare them minutely, part by part. Until that is done things that are really very dissimilar often appear absolutely alike. Two lines of very different lengths will appear about equal when lying in different directions; but if we put them parallel with their distant ends even, and then look at the nearer ends, we can directly perceive their inequality.

So it’s not always as easy as you might think to ascertain •whether two phenomena are alike and •how they differ if they do differ. When the two can’t be brought together in a way that lets the observer compare their several parts in detail, he must come at the comparison indirectly, through reasoning and general propositions. When we can’t bring two straight lines together to determine whether they are equal ·in length·, we do it with the •physical aid of a foot-rule applied first to one and then to the other, and the •logical aid of the general proposition ‘Things that are equal to the same thing are equal to one another’. The comparison of two things through the intervention of a third thing when their direct comparison is impossible—that’s the appropriate scientific process for ascertaining resemblances and dissimilarities, and it’s the sum total of what logic has to teach on this subject.

Locke stretched this line of thought too far, holding that •reasoning itself is nothing but the comparison of two ideas through the medium of a third, and •knowledge is the perception of the agreement or disagreement of two ideas—doctrines that the Condillac school blindly adopted, without the qualifications and distinctions that they were carefully guarded with by their illustrious author. Of course when the question one is pursuing is actually about the agreement or disagreement (i.e. the resemblance or dissimilarity) of two things, as happens especially in the arithmetic and geometry, then if a solution can’t be found by direct perception it must be indirectly sought by comparing these two things through the medium of a third. But this is far from being true of all inquiries. The knowledge that bodies fall to the ground is a perception not of •agreement or disagreement but of •a series of physical occurrences, a succession of sensations. Locke’s definitions of knowledge and of reasoning needed to be limited to knowledge of and reasoning about resemblances. Even then, what he says isn’t strictly correct, because the comparison is made not between ‘the ideas of’ the two phenomena but between the phenomena themselves. I pointed out this mistake in I.5.1 and II.5.5, and traced it to an imperfect conception of what happens in mathematics, where very often the comparison really is made between the ideas, without any appeal to the outward senses; but that’s only because in mathematics a comparison of the ideas is strictly equivalent to a comparison of the phenomena themselves. In the case of numbers, lines, and figures, our idea of an object is a complete picture of the object so far as the matter in hand is concerned; so we can learn from the picture whatever could be learned from the object itself by merely contemplating it at the instant when the picture is taken. No mere contemplation of •gunpowder would ever teach us that a spark would make it explode, so the contemplation of •the idea of gunpowder wouldn’t do that either; but the mere contemplation of a straight line shows that it can’t enclose a space, so the contemplation of the idea of it will show the same. What takes place in mathematics is thus no argument that the comparison is always between the ideas. It is always, either indirectly or directly, a comparison of the phenomena.

In some cases we can’t bring the phenomena to the test of direct inspection at all, or not in a precise enough way, but must judge of their resemblance by inference from other (dis)similarities that are more open to observation. In those cases we of course require, as in all ratiocination, generalisations or formulae applicable to the subject. We must reason from laws of nature—from observable uniformities involving likeness or unlikeness.

§3. The most comprehensive of these laws or uniformities are the ones supplied by mathematics—the axioms relating to equality, inequality, and proportionality, and the various theorems based on them. And these are the only Laws of Resemblance that need to be treated separately—indeed the only ones that can. There are indeed countless other theorems affirming resemblances among phenomena, e.g. that the angle of the reflection of light is equal to its angle of incidence (equality being merely exact resemblance in magnitude), and that the planets describe equal areas in equal times. . . . Propositions like these affirm resemblances of the same sort as those asserted in mathematical theorems; what is different ·between mathematics and physical sciences· is that the propositions of mathematics are true of all phenomena, or at least without distinction of origin; while the truths of physical science are affirmed only of special phenomena that originate in a certain way; and the equalities, proportionalities, or other resemblances that exist between such phenomena must be either derived from, or identical with, the law of their origin—the law of causation they depend on. The equality of the areas described in equal times by the planets is derived from the laws of the causes, and until its derivation was shown it was ·merely· an empirical law. The equality of the angles of reflection and incidence is identical with the law of the cause; because the cause is a light-ray’s hitting a reflecting surface, and the equality in question is the very law according to which that cause produces its effects. So this class of uniformities of resemblance between phenomena are inseparable—in fact and in thought—from the laws of the production of those phenomena; and the principles of induction applicable to them are precisely the ones I have discussed in the preceding chapters of this Book.

Not so with mathematical truths. The laws of equality and inequality between spaces, or between numbers, have no connection with laws of causation. The proposition that

the angle of reflection equals the angle of incidence

is a statement of the mode of action of a particular cause; the proposition but that

when two straight lines intersect, the opposite angles are equal

is true of all such lines and angles, whatever their causes are. That

the squares of the periodic times of the planets are proportional to the cubes of their distances from the sun

is a uniformity derived from the laws of the causes (or forces) that produce the planetary motions; but that

the square of any number is four times the square of half the number

is true independently of any cause. So the only laws of resemblance that we have to consider independently of causation are those of mathematics.

§4. The same thing is evident with respect to the last of the five categories ·listed here·, namely order in place. The order in place of the effects of a cause x is (like everything else that’s true of x’s effects) a consequence of x’s laws. The order in place—which I have been calling the ‘collocation’—of the absolutely basic causes is. . . .in each instance an ultimate fact in which no laws or uniformities are traceable. The only remaining general propositions about order in place, and the only ones having nothing to do with causation, are some of the truths of geometry. I’m talking about laws that enable us to infer from •the order in place of certain points, lines, or spaces •the order in place of others that are connected with the former in some known way, this being done without bringing in the physical cause from which they happen to derive their origin, and indeed without bringing in any facts about those points, lines, or spaces other than facts about position or magnitude.

It turns out, therefore, that mathematics is the only department of science whose methods I still have to inquire into. This needn’t take long, because I have already gone a fair distance into it in Book II. I said there that the directly inductive truths of mathematics are few in number—only •the axioms and •certain existence-propositions that are tacitly involved in most of the so-called definitions. And I gave reasons—seemingly conclusive ones—for affirming that these basic premises from which the remaining truths of mathematics are deduced are results of observation and experience, i.e. are based on the evidence of the senses. They don’t seem to be, but they are. That things equal to the same thing are equal to one another and that two straight lines that have once intersected one another continue to diverge are inductive truths that rest—as does the law of universal causation—only on induction per enumerationem simplicem, i.e. on the fact that they have been perpetually perceived to be true and never once found to be false. But ·there’s a difference between the law of causation and these mathematical axioms·. For a long time there were events that appeared not to be caused, though really they were; but with the axioms of mathematics there aren’t even apparent exceptions. All that’s needed to perceive the truth of one of them in any individual case is the simple act of looking at the objects in a proper position. Their infallible truth was recognised from the very dawn of theoretical thought; and because their extreme familiarity made it impossible for the mind to conceive the objects in any other way, the axioms came to be (and still are) generally considered as self-evidently true, i.e. as truths recognised by instinct.

§5. Something that seems to require explanation is the fact that the immense multitude of truths in the mathematical sciences (a multitude still as far as ever from being exhausted) can be extracted from so few elementary laws. It’s hard to see, at first, how there can be room for such an infinite variety of true propositions on subjects that are apparently so limited.

To begin with the science of number. The elementary or ultimate truths of this science are the common axioms concerning equality, namely, ‘Things that are equal to the same thing are equal to one another’, and ‘Equals added to equals make equal sums’ (no other axioms are required[1]), together with the definitions of the various numbers. Like other so-called definitions, these are composed of two things—•the explanation of a name and •the assertion of a fact; and only the latter of these can be a first principle or premise of a science. The fact asserted in the definition of a number is a physical fact. Each of the numbers two, three, four, etc. denotes physical phenomena, and connotes [see Glossary] a physical property of those phenomena. Two denotes all pairs of things, and twelve denotes all dozens of things, connoting what makes them pairs or dozens; and what makes them so is something physical—because it can’t be denied that two apples are physically distinguishable from three apples, two horses from one horse, and so on, i.e. that they are a different visible and tangible phenomenon. I’m not undertaking to say what the difference is; it is enough that there is a difference that the senses can recognise. And although a 102 horses are not distinguished from 103 horses as easily as two horses are from three, the horses can be so placed that the difference is perceptible; if that weren’t so we would never have distinguished them and given them different names. Everyone knows that weight is a physical property of things; yet small differences between great weights are as imperceptible to the senses in most situations as small differences between great numbers. They become evident only when the two objects are placed in a special position—namely, in the opposite scales of a delicate balance.

Well, then, what is connoted by a name of number—·i.e. by a numeral·? Of course it’s some property belonging to the agglomeration of things that we call by the name; and that property is

the characteristic manner in which the agglomeration is made up parts and can be separated into parts.

I’ll try to make this more intelligible by a few explanations.

When we call a collection of objects ‘two’, ‘three’, or ‘four’, they aren’t two, three, or four in the abstract; they are two, three, or four things of some particular kind—pebbles, horses, inches, pounds’ weight. What the numeral [see Glossary] connotes is the way single objects of the given kind must be put together to produce that particular aggregate. If it’s an aggregate of pebbles, and we call it ‘two’, the name implies that to compose the aggregate one pebble must be joined to one pebble. If we call it ‘three’, one and one and one pebble must be brought together to produce it, or else one pebble must be joined to an already existing aggregate of the kind called ‘two’. The aggregate that we call ‘four’ has a still greater number of characteristic modes of formation. One and one and one and one pebble may be brought together; or two aggregates of the kind called ‘two’ may be united; or one pebble may be added to an aggregate of the kind called ‘three’. Every number in the ascending series can be formed by joining smaller numbers in a growing variety of ways. Even limiting the parts to two, the number can be formed (or divided) in as many different ways as there are numbers smaller than itself; and there are even more ways of doing it if we admit threes, fours, etc. Other ways of reaching the same aggregate present themselves, not by uniting smaller aggregates but dismembering larger ones: three pebbles can be formed by removing one pebble from an aggregate of four, two pebbles by an equal division of a similar aggregate, and so on.

Every arithmetical proposition—every statement of the result of an arithmetical operation—is a statement of one of the ways of forming a given number. It affirms that •a certain aggregate A could have been formed by putting together certain other aggregates, or by removing certain portions of some aggregate, and that •and that therefore we could reproduce those aggregates from A by reversing the process.

Thus, when we say that 12^3 = 1728, what we affirm is this:

If having a sufficient number of pebbles (say), we put them together into the particular sort of aggregates called ‘twelves’, and put together these twelves again into similar collections, and finally make up twelve of these largest parcels, the aggregate we have formed will be of the sort we call ‘1728’—namely, that which (to take the most familiar of its modes of formation) can be made by joining the parcel called ‘a thousand’ pebbles, the parcel called ‘seven hundred’ pebbles, the parcel called ‘twenty’ pebbles, and the parcel called ‘eight’ pebbles.

The converse proposition that the 1728−3 = 12 says that this large aggregate can again be decomposed into the twelve twelves of twelves of pebbles that it consists of.

There are countless ways of forming any number; but when we know one way of forming a number, all the other ways can be determined deductively. If we know that

•a is formed from b and c,

•b is formed from d and e,

•c is formed from d and f,

and so forth, until we have included all the numbers of any scale we choose to select, we have a set of propositions from which we can reason to all the other ways of forming those numbers from one another. (In doing this we must take care that for each number the mode of formation is really a distinct one, not bringing us round again to the former numbers but introducing a new one.) Having established a chain of inductive truths connecting all the numbers of the scale, we can ascertain the formation of any one of those numbers from any other merely by travelling from one to the other along the chain. Suppose that we know only the following modes of formation:

6 = 4 + 2

4 = 7 − 3

7 = 5 + 2

5 = 9 − 4.

We could determine how 6 can be formed from 9. For 6 = 4 + 2 = 7 − 3 + 2 = 5 + 2 − 3 + 2 = 9 − 4 + 2 − 3 + 2. So it can be formed by taking away 4 and 3, and adding 2 and 2. If we also know that 4 = 2 + 2, we can get 6 from 9 by merely taking away 3.

So we need only to select one of the various ways of forming each number, and then we can ascertain all the rest. And since the understanding finds easiest to receive and retain things that are uniform and therefore simple, there’s an obvious advantage in •selecting a number-forming mode that is alike for all, •fixing the connotation of numerals on one uniform principle. The system of numerals that we actually use has this advantage, and the additional one of conveying to the mind two of the ways of forming every number. Each number is regarded as formed by adding a unit to the number next below it, and this way of forming it is conveyed by its place in the series. And each is also regarded as formed by adding a number of units less than ten, and a number of aggregates each equal to one of the successive powers of ten; and this way of forming it is expressed by its spoken name and by its numerical character.

What makes arithmetic the type [see Glossary] of a deductive science is the role in it of the comprehensive law ‘The sums of equals are equals’ or (in language that is less familiar but theoretically better) ‘Whatever is made up of parts is made up of the parts of those parts’. This truth is obvious to •the senses in all cases that it makes sense to submit to •their judgment, and is so general that it’s coextensive with nature itself; and because it’s true of all sorts of phenomena. . . .it must be considered an inductive truth—or law of nature—of the highest order. Every arithmetical operation is an application of this law or of other laws that can be deduced from it. This is our warrant for all calculations. We believe that 5 + 2 = 7 on the strength of •this inductive law combined with •the definitions of those numerals. We arrive at that conclusion—as you may remember from your childhood—by adding units one at a time: 5 + 1 = 6, therefore 5 + 1 + 1 = 6 + 1 = 7 and again 2 = 1 + 1, therefore 5 + 2 = 5 + 1 + 1 = 7.

§6. The countless true propositions about particular numbers can’t unaided give an adequate conception of the extent of the truths that make up the science of number. The propositions I have been speaking of are the least general of all numerical truths. It’s true that even these are coextensive with all nature; the properties of the number four are true of anything that is divisible into four equal parts, and everything is so divisible either actually or ideally. But the propositions making up the science of algebra are true not ·merely· of •a particular number but of •all numbers; not ·merely· of all things considered •as being divided in a particular way but of all things considered •as being divided in any way—as being designated by a numeral at all.

Any number’s mode of formation belongs to it alone; it couldn’t also be the mode of formation of some other number; so it’s a kind of paradox to say both that

all propositions that can be made about numbers relate to how they are formed from other numbers

and yet that

some propositions are true of all numbers.

But this very paradox leads to the real source of generalisation about the properties of numbers. Two numbers can’t be formed in the same way from the same numbers; but they can be formed in the same way from different numbers—as nine is formed from three by multiplying it into itself, and sixteen is formed from four by the same process. Thus there arises a classification of ways of forming numbers—i.e. (in the language mathematicians prefer) a classification of functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are ways of forming numbers. There aren’t many simple functions. Most functions are formed by combining several of the operations that form simple functions, or by repetitions of one of those operations. The simple functions of any number x are all reducible to the following forms:

x + a

x − a

ax

x ⁄ a

xa

a√x

log x (to the base of a)

and the same expressions varied by switching x and a wherever that switch would alter the value. . . . All other functions of x are formed by putting some one or more of the simple functions in the place of x or a, and subjecting them to the same elementary operations.

In order to reason generally about functions we need a system of naming that enables us to express any two numbers by names that show what function each is of the other, without saying what particular numbers they are. . . . The system of general language called ‘algebraical notation’ does this. The expressions a and a^2 + 3a denote, respectively, •any number and •the number formed from that in a particular way. The expressions a, b, n, and (a + b)n denote •any three numbers and •a fourth that is formed from them in a certain way.

Here is the general problem of the algebraical calculus: F being a certain function of a given •number, find what function F will be of any •function of that number. For example, a binomial a + b is a function of its two parts a and b, and the parts are in their turn functions of a + b. Now, (a + b)n is a certain function of the binomial; what function will this be of a and b, the two parts? The answer is the binomial theorem. [Mill states the theorem in its general form; it’s hard to take in, and for present purposes it may be enough to say that the special case of it where n = 2 is the familiar equation

(a + b)2 = a^2 + 2ab + b^2

and where n = 3

(a + b)3 = a^3 + 3a^2b + 3ab^2 + b^3

and so on. Mill continues:] This shows how the number that is formed by multiplying a + b into itself n times could be formed without that process, directly from a, b, and n. All the theorems of the science of number are like that. They assert the identity of the result of different ways of forming numbers. They affirm that some process of number-forming from x produces the same number as some process of number-forming from a certain function of x.

Besides these general truths or formulae, what remains in the algebraical calculus is the resolution of equations. But the resolution of an equation is also a theorem. If the equation is

x^2 + ax = b

the resolution of it, namely

x = ½a ± √¼a^2 + b

is a general proposition, which may be regarded as an answer to the question: ‘If b is a certain function of x and a—namely x^2 + ax—what function is x of b and a? The resolution of equations is, therefore, a mere variety of the general problem as I have stated it. The problem is: Given a function, what function is it of some other function? And in the resolution of an equation, the problem is to find what function of one of its own functions the number itself is.

That tells you what algebra aims to do. As for its ways of doing it, everyone knows that they are simply deductive. In demonstrating a theorem or solving an equation we travel from the datum to the quaesitum [= ‘from the given to the sought’ = ‘from the problem to the solution’] by pure ratiocination. The only premises are •the original hypothesis ·or problem or equation to be solved· and •the fundamental axioms that things equal to the same thing are equal to one another, and that the sums of equal things are equal. At each step in the demonstration or in the calculation, we apply one or other of these truths or truths deducible from them. . . .

This isn’t the place to go further into the analysis of the truths and processes of algebra. There’s also no need for me to do so, because a great deal of the task has been performed by other writers. . . . The profound treatises of a truly philosophical mathematician, Augustus De Morgan, should be studied by everyone who wants to understand •why mathematical truths are evident, and •what is meant by the more obscure processes of algebra. What August Comte writes in his Cours de Philosophie Positive about the philosophy of the higher branches of mathematics is among the many valuable gifts for which philosophy is indebted to that eminent thinker.

§7. The extreme generality of the laws of number, and their remoteness. . . .from visual and tactual imagination, makes it rather difficult. . . .to think of them as really being physical truths obtained by observation. But that difficulty doesn’t arise with regard to the laws of extension. The facts expressed by those laws are of a kind specially accessible to the senses, and suggesting admirably clear images to the imagination. That geometry is a strictly physical science would doubtless have been recognised down through the centuries if it hadn’t been for the illusions produced by two circumstances: (i) the fact (which I mentioned earlier) that the truths of geometry can be collected from our ideas or mental pictures of objects as effectively as from the objects themselves; and (ii) the demonstrative nature of geometrical truths, which at one time was supposed to constitute a deep difference between them and physical truths, the latter resting on merely probable evidence and therefore regarded as essentially uncertain and imprecise. The advance of knowledge, however, has shown plainly that physical science in its better understood branches is quite as demonstrative as geometry. The task of deducing its details from a few comparatively simple principles turns out to be anything but the impossibility it was once thought to be; and the supposed greater certainty of geometry is an illusion, arising from the ancient prejudice which mistakes the ideal data from which we reason •in geometry for a special class of realities, while the corresponding ideal data in any deductive •physical science are recognised as what they really are, hypotheses.

Every theorem in geometry is a law of external nature, and could have been discovered by generalising from observation and experiment, which in this case come down to comparison and measurement. But it was found to be convenient and therefore desirable to deduce these truths by ratiocination from a small number of general laws of nature—the first principles and basic premises of the science—whose certainty and universality are obvious to the most casual observer. Among these general laws must be included the two that I have presented as basic principles of the science of number also, and are applicable to every sort of quantity. I mean

•The sums of equals are equal, and

•Things that are equal to the same thing are equal to one another;

the latter of which can be expressed in a way that more openly suggests the inexhaustible multitude of its consequences, namely:

•Whatever is equal to any one of a number of equal magnitudes, is equal to any other of them.

For geometry we must add a third law of equality, namely:

•Lines, surfaces, and solid spaces that can be applied to one another so that they coincide are equal.

Some writers have said that this law of nature is a mere verbal definition, that ‘equal magnitudes’ means nothing but ‘magnitudes that can be applied to one another so that they coincide’. I don’t agree. The equality of two geometrical magnitudes can’t differ fundamentally in its nature from the equality of two weights, two degrees of heat, or two stretches of time, and the proposed definition of equality isn’t suitable for any of these. None of these things can be ‘applied to one another so that they coincide’, yet we understand perfectly what we mean by calling them ‘equal’. Things are equal in magnitude, as in weight, when they are felt [Mill’s word] to be exactly similar in respect of the attribute in which we compare them. As for the application of lines etc. to each other in geometry, that’s merely bringing them into a position in which our senses can recognise deficiencies of exact resemblance that would otherwise escape our notice. It’s on a par with balancing objects in a pair of scales to determine whether their weights are equal.

Along with these three general principles or axioms, the other premises of geometry are the so-called definitions—i.e. propositions each of which •asserts the real existence of some object and •states some one property of it. In some cases more than one property is commonly assumed, but there’s never a need for more than one. It is assumed that there are such things in nature as straight lines, and that any two of them setting out from the same point diverge more and more without limit. This assumption (which includes and goes beyond Euclid’s axiom that two straight lines can’t enclose a space) is as indispensable as any of the other axioms in geometry, and it’s as evident as they are because like them it rests on a simple, familiar, and universal observation. It is also assumed that straight lines diverge from one another in different degrees, meaning that there are such things as angles and that they can be equal or unequal. It’s assumed that there is such a thing as a circle, and that all its radii are equal; such things as ellipses, and that the sums of the focal distances are equal for every point in an ellipse; such things as parallel lines, and that those lines are everywhere equally distant.[2]

§8. It is a matter of more than curiosity to ask:

What special feature of the physical truths that are the subject of geometry makes them all deducible from such a small number of original premises? Why it is that we can start with •one characteristic property of each kind of phenomenon and •two or three general truths relating to equality, and travel from mark to mark until we obtain a vast body of derivative truths that don’t look a bit like those elementary ones?

The explanation of this remarkable fact seems to lie in the following ·two· facts. First, all questions of position and figure can be resolved into questions of magnitude. The position and figure of any object are determined by determining the position of a sufficient number of points in it; and the position of any point can be determined by the magnitude. . . .of the perpendiculars drawn from the point to three planes at right angles to one another, arbitrarily selected. This transformation of all questions of quality into questions only of quantity turns geometry into the single problem of the measurement of magnitudes, i.e. the ascertaining of the equalities between them. Now remember that ascertaining any equality between x and y

•proves (according to one of the general axioms) as many other equalities as there are other things equal to either x or y, and that

•proves (according to another of the axioms) the equality of as many pairs of magnitudes as can be formed by the numerous operations that resolve themselves into the addition of x and y to one another or to other equals.

When we bear that in mind, we cease to be puzzled by the fact that •the more a science has to do with equality the more copious its supply of marks of marks, and that •the sciences of number and extension, which have to do with equality and little else, are the most deductive of all the sciences.

Secondly, two or three of the principal laws of space or extension are especially well fitted for making one position or magnitude a mark of another, thereby contributing to making the science largely deductive. •The magnitudes of enclosed spaces, whether in two or three dimensions, are completely determined by the magnitudes of the lines and angles that bound them. •The length of any line, straight or curved, is measured (certain other things being given) by the angle it subtends, and vice versa. •The angle that any two straight lines make with each other at an inaccessible point is measured by the angles they separately make with any third line we choose to select. By means of these general laws, the measurement of all lines, angles, and spaces could be accomplished by measuring a single straight line and a large enough number of angles—which is what they actually do in making a trigonometrical survey of a country. It’s lucky for us that this is practicable, because the exact measurement of long straight lines is always difficult and often impossible, whereas angles are easy to measure. Those three generalisations provide such facilities for indirectly measuring magnitudes (by supplying us with known lines or angles that are marks of the magnitude of unknown ones, and thereby of the spaces they enclose), that it’s easy to understand how from a few data we can go on to ascertain the magnitude of indefinitely many lines, angles, and spaces that we couldn’t easily measure—or couldn’t measure at all—by any more direct process.

§9. I have said all I need to say here about the laws of nature that are the special subject of the sciences of number and extension. The immense part those laws play in giving a deductive character to the other branches of physical science is well known; and it’s not surprising, when we consider that all causes operate according to mathematical laws. The effect is always dependent on—i.e. is a function of—the cause’s quantity and generally of its position also. So we can’t reason about causation without introducing considerations of quantity and extension at every step; and when the phenomena are such that we can get accurate enough numerical data, the laws of quantity become the grand instrument for calculating forward to an effect or backward to a cause. In all other sciences, as well as in geometry, questions of quality nearly always depend on questions of quantity, as can be seen in the most familiar phenomena, even colour. When a painter mixes colours on his palette, the comparative quantity of each entirely determines the colour of the mixture.

[For further discussion of these matters Mill refers the reader to Comte’s Cours de Philosophie Positive, which he also credits with a full discussion of Mill’s next topic, namely:] the limits to how far mathematical principles can be used to improve other sciences. They obviously can’t be used on classes of phenomena whose causes

•are so little open to our observation that we can’t ascertain their numerical laws by a proper induction; or

•are so numerous and intermixed in such a complex way that even if their laws were known the computation of the over-all effect is beyond the powers of mathematics as it is or is likely to be; or

•are themselves are in a state of perpetual fluctuation— as in physiology, and still more (if possible) in the social sciences.

The mathematical solutions of physical questions become progressively more difficult and imperfect in proportion as the questions lose their abstract and hypothetical character and come closer to the degree of complication actually existing in nature. [The quotations that follow are from Comte.] The result is that except for astronomical phenomena and those most nearly analogous to them, mathematical accuracy is generally obtained ‘at the expense of the reality of the inquiry’; while even in astronomical questions, ‘despite the admirable simplicity of their mathematical elements, our feeble intelligence becomes incapable of effectively following out the logical combinations of the laws on which the phenomena depend, as soon as we try to take into consideration more than two or three essential influences at once’. A remarkable example of this is the three-body problem that I mentioned here—a comparatively simple question the complete solution of which has defeated the skill of the most profound mathematicians. This shows us that mathematical principles can’t be usefully applied to phenomena that depend on the mutual action of the innumerable minute particles of bodies, e.g. •chemistry, and still more •physiology. And for similar reasons those principles remain inapplicable to the still more complex inquiries into the phenomena of •society and government.

The value of mathematical instruction as a preparation for those more difficult investigations consists in the applicability not of its •doctrines but of its •method. Mathematics will always be the most perfect type of the deductive method in general; and the applications of mathematics to the deductive branches of physics provide the only classroom in which philosophers can effectively learn the most difficult and important part of their art, namely the use of the laws of simpler phenomena for explaining and predicting the laws of more complex ones. These grounds are quite sufficient for regarding mathematical training as an indispensable basis of real scientific education, and regarding (according to the dictum which an old but unauthentic tradition ascribes to Plato) one who is ignorant of mathematics as lacking in one of the most essential qualifications for successfully pursuing the higher branches of philosophy.

NOTES

[1] The axiom, ‘Equals subtracted from equals leave equal differences’ can be demonstrated from the two axioms in the text. If A = a and B = b, then A−B =a−b. For if not, let A−B =a−b+c. Then since B =b, adding equals to equals A=a+c. But A=a. Therefore a=a+c, which is impossible.

—This proposition having been demonstrated, we can use it to demonstrate the following: ‘If equals are added to unequals, the sums are unequal.’ If A = a and B 6= b, A+B 6= a+b. For suppose A+B = a+b. Then, since A = a and A+B = a+b, subtracting equals from equals, B = b; which is contrary to the hypothesis.

—We can also prove ‘Two things of which one is equal and the other unequal to a third thing are unequal to one another’. If A = a and A 6= B, then a6=B. For suppose a=B. Then since A=a and a=B, and since things equal to the same thing are equal to one another, A=B; which is contrary to the hypothesis.

[2] Geometers have usually preferred to define parallel lines by the property of being in the same plane and never meeting. But this has required them to assume as an additional axiom some other property of parallel lines; and the unsatisfactory way in which Euclid and others have selected properties for that purpose by has always been regarded as the disgrace of elementary geometry. Equidistance is a fitter property to characterise parallels by, even as a verbal definition, because it is the attribute really involved in the name’s meaning. If all that is meant by ‘x any y are parallel’ were ‘x and y are in the same plane and never meet’, we would happily speak of a curve as ‘parallel to’ its asymptote [i.e. to a line that gets nearer to it ad infinitum but doesn’t meet it]. The meaning of ‘parallel lines’ is ‘lines that run in exactly the same direction and therefore don’t become nearer or further from one another’—a conception immediately suggested by the contemplation of nature. That the lines •will never meet is of course included in the more comprehensive proposition that they •are everywhere equally distant. And that any straight lines that are in the same plane and not equidistant will certainly meet can be demonstrated in the most rigorous manner from the basic property of straight lines assumed in the text, namely that if they set out from the same point they diverge more and more without limit.

Chapter 25. The grounds of disbelief

§1. In the past 24 chapters I have discussed—as far as space and my abilities permitted—the method of arriving at general truths (i.e. general propositions fit to be believed) and the nature of the evidence they are based on. But an examination of evidence doesn’t always produce belief, or even suspension of judgment; it sometimes produces disbelief. So a complete philosophy of induction and experimental inquiry must treat the grounds not only of belief but also of disbelief. I’ll devote my final chapter to that.

By ‘disbelief’ I don’t mere absence of belief. The ground for abstaining from belief is simply the absence or insufficiency of proof; and in considering what is sufficient evidence to support a conclusion I have already implicitly considered what evidence is not sufficient for the same purpose. By ‘disbelief’ I mean the state of mind in which we are fully convinced that some opinion is not true; so that if evidence— even apparently strong evidence—were produced in favour of the opinion, we would believe that the witnesses spoke falsely, or that they or we ourselves (if we were the direct percipients) were mistaken.

No-one is likely to deny that there are such cases. Assertions for which there is abundant positive evidence are often disbelieved because of what is called their ‘improbability’ or ‘impossibility’. The question we have to think about is: ‘What do those two words mean in this context? And how far and in what circumstances do the properties they express give sufficient grounds for disbelief?’

§2. When positive evidence produced in support of an assertion is rejected because it is impossible or improbable, it never amounts to full proof. It is always based on some approximate generalisation. The claim may have been asserted by a hundred witnesses, but the thesis that whatever a hundred witnesses affirm is true has many exceptions. We may seem to ourselves to have actually seen the fact, but the thesis that we really see what we think we see is far from being a universal truth—our sense-organs may have been diseased, or we may have •inferred something and imagined that we •perceived it. Thus, given that the evidence for the affirmative is never more than an approximate generalisation, everything will depend on what the evidence is for the negative. If that also rests on an approximate generalisation, this is a case for comparison of probabilities. If the approximate generalisations leading to the affirmative add up to something less strong—i.e. further from being universal—than the approximate generalisations that support the negative side of the question, the proposition is said to be ‘improbable’ and is to be disbelieved provisionally. But when an alleged fact contradicts (not any number of approximate generalisations, but) a completed generalisation based on a rigorous induction, it is said to be ‘impossible’ and is to be disbelieved totally [here = ‘unconditionally’].

This last principle, simple and evident as it appears, aroused a violent controversy on the occasion of an attempt to apply it to the question of the credibility of miracles. Hume’s celebrated doctrine that nothing is credible that is contradictory to experience or at variance with laws of nature is merely the plain and harmless proposition that whatever is contradictory to a complete induction is incredible. That such a maxim as this should be accounted •a dangerous heresy or •a great and recondite truth speaks ill for the state of philosophical theorising on such subjects!

You may want to ask:

Doesn’t the very statement of the proposition imply a contradiction? An alleged fact, according to this theory, is not to be believed if it contradicts a complete induction. But a complete induction mustn’t contradict any known fact. So isn’t it a petitio principii [see Glossary] to say that the fact ought to be disbelieved because the induction opposed to it is complete? How can we have a right to declare the induction complete when facts supported by credible evidence present themselves in opposition to it?

We do have that right whenever the scientific canons of induction give it to us, i.e. whenever the induction can be complete. We have it, for example, in a case of causation where there has been a decisive experiment. If A is added to a set of antecedents that hasn’t been followed by B, and B does now follow, then in that instance A is B’s cause or an indispensable part of its cause; and if A is tried again with many different sets of antecedents and B still follows, then it is the whole cause. (In each case it is of course essential that adding A to a set of antecedents doesn’t change the set in any other way.) If these observations or experiments are repeated often enough, and by enough people, to exclude any suspicion of error in the observer, a law of nature is established; and as long as this law is accepted as such, the assertion that on some particular occasion

A occurred and B didn’t follow, though there was no counteracting cause

must be disbelieved. Such an assertion shouldn’t be credited on any evidence short of what would suffice to overturn the law. The general truths that

•Whatever has a beginning has a cause, and

•When none but the same causes exist, the same effects follow,

rest on the strongest inductive evidence possible; whereas the proposition that

•Things affirmed by a crowd of respectable witnesses are true

is only an approximate generalisation; and—even if we fancy we actually saw or felt whatever-it-was that contradicts the law—what a human being can see is merely a set of appearances, from which the real nature of the phenomenon is merely an inference, and such inferences usually make heavy use of approximate generalisations. So if we decide to hold by the law, no amount of evidence ought to persuade us that something that contradicts it has happened. If the evidence E that is produced makes it more likely that •the observations and experiments the law is based on were inaccurately performed or incorrectly interpreted than that •E is false, we may believe the evidence; but then we must abandon the law. And since the law had been accepted on the basis of what seemed to be a complete induction, it can only be rejected on evidence equivalent to that—i.e. as being inconsistent not with •any number of approximate generalisations but with •some other and better established law of nature. The extreme case of a conflict between two supposed laws of nature has probably never actually occurred in contexts where each ‘law’ was investigated according to the true canons of scientific induction; but if it did occur, it would have to lead to the total rejection of one of the ‘laws’. It would prove that there’s a flaw in the logical process by which one or other of the ‘laws’ was established, showing that that supposed general truth is no truth at all. We can’t admit a proposition as a law of nature while believing something that contradicts it. We must disbelieve the alleged fact, or believe that we were mistaken in accepting the supposed law.

For an alleged fact to contradict a law of causation, the allegation must be. . . .that this happened in the absence of any adequate counteracting cause. Now, in the case of an alleged miracle, the assertion is the exact opposite of this. It is that the effect was defeated not in •the absence of a counteracting cause but in •consequence of one, namely, an. . . .act of the will of some being who has power over nature; and in particular of a Being whose will is assumed to have given all the causes their causal powers and can therefore easily be supposed to be able to counteract them. As Thomas Brown rightly said in his Inquiry into the Relation of Cause and Effect, a miracle doesn’t contradict the law of cause and effect; it is a •new effect that is supposed to be produced by the introduction of a •new cause. There can be no doubt that this cause, if present, is adequate to do the job; the only antecedent improbability that can be ascribed to the miracle is the improbability that any such cause exists.

So all that Hume has shown—and this he must be credited with showing—that no evidence can prove a miracle to anyone who

•doesn’t already believe in the existence of one or more beings with supernatural power; or

•believes he has full proof that the character of the Being whom he recognises is inconsistent with His having interfered on the occasion in question.

[Mill builds into his statement of what Hume showed the proviso ‘at least in the imperfect state of our knowledge of natural agencies, which leaves it always possible that some of the physical antecedents may have been hidden from us’. It’s not obvious how this fits in, and Mill doesn’t explain it.]

If we don’t already believe in supernatural agencies, no miracle can prove their existence to us. That the supposed miracle actually occurred, considered merely as an extraordinary fact, can be satisfactorily certified by our senses or by testimony; but nothing can ever prove that it was a miracle, because there’s always the rival hypothesis that it was a result of some unknown natural cause; and this possibility can’t be shut out so completely that the only alternative remaining is to admit the existence and intervention of a being superior to nature. Those who already believe in such a being have two hypotheses to choose from, a •supernatural agency and an unknown •natural agency, and they have to judge which of the two is more probable in the particular case. In working towards a judgment about this they’ll have to think about whether it would be in character for the Deity, as they conceive him, to have caused this particular event. But with the knowledge we now have of the general uniformity of the course of nature, religion has been compelled to follow in the wake of science by acknowledging that the over-all government of the universe is carried on by general laws and not by special interpositions. For anyone who holds this belief there’s a general presumption against any supposition of divine agency not operating through general laws. In other words, for such a person there’s an antecedent improbability in every miracle—an improbability that could be outweighed only by an extraordinarily strong antecedent probability based on the special features of the case.

§3. So the assertion that a cause has failed to produce an effect that is connected with it by a completely ascertained law of causation is to be disbelieved or not according to the probability or improbability that this particular instance contained an adequate counteracting cause. To estimate this isn’t harder than estimating other probabilities. With regard to all known causes that could counteract the given causes we usually have some previous knowledge of how often they occur, from which we can infer the antecedent improbability of their having been present in any particular case. And with known or unknown causes we don’t have to pronounce on the probability of their existing in nature, but only the probability of their having existed at the time and place at which the miracle is alleged to have happened. We usually have the means (when the circumstances of the case are at all known to us) of judging how likely it is that such a cause existed at that time and place •without showing its presence by some other marks and (in the case of an unknown cause) •without having shown its existence ever before. . . .

So much for the case where the alleged fact conflicts, or appears to conflict, with a real law of causation. A more common case, perhaps, is that of its conflicting with •mere uniformities of coexistence that aren’t proved to depend on causation, i.e. with •the properties of Kinds. It is with these uniformities that travelers’ marvellous stories are apt to conflict—e.g. tales of men with tails or with wings, and (until confirmed by experience) of flying fish; or of ice, in the famous anecdote of the Dutch travelers and the King of Siam. Facts of this description—facts that haven’t previously been heard of, but that no known law of causation implies to be impossible, are what Hume characterises as not •contrary to experience but merely •unconformable to it. . . .

In a case of this sort, the fact asserted is the existence of a new Kind. This in itself is not in the least incredible, and should be rejected only if the improbability

that any sort of object existing at that particular place and time should have gone undiscovered until now

is greater than the improbability

that the witnesses were mistaken or lied.

Accordingly, when such assertions are made by credible persons and concern unexplored places, they aren’t disbelieved but only regarded as requiring confirmation from subsequent observers—unless the alleged properties of the supposed new Kind conflict with known properties of some larger Kind that includes it. . . .as in the case of Pliny’s men, or any other kind of animal with a structure different from what has always been found to coexist with animal life. As for how to deal such a case, I needn’t add much to what I said in chapter 22 (here). When the uniformities of coexistence that the alleged fact would violate are such as to raise a strong presumption of their being the result of causation, the fact that conflicts with them should be disbelieved—at least provisionally, subject to further investigation. When the presumption amounts to a virtual certainty, as with the general structure of organisms, all we have to ask is this: ‘In phenomena as little understood as this. . .

. . . mightn’t there be at work a counteracting cause that we haven’t known about before? or

. . . mightn’t the phenomena be capable of originating in some other way that would produce a different set of derivative uniformities?’

In some cases neither of those suppositions can be regarded as very improbable, because the generalisation to which the alleged fact would be an exception is very special and of limited range. Examples are the reports about •flying fish and about •the ornithorhynchus [= the platypus, an egg-laying, venomous, duck-billed, beaver-tailed, otter-footed mammal found only in Australia]. Faced with reports of such alleged anomalies, it is wise to suspend our judgment pending the subsequent inquiries that are sure to confirm the assertion if it is true. But when the generalisation is very comprehensive, taking in a vast number and variety of observations and covering a considerable province of nature’s domain, then for reasons that I have fully explained such an •empirical law comes near to the certainty of an ascertained •law of causation; and alleged exceptions to it ought not to be accepted except on the evidence of some law of causation that is proved by a still more complete induction.

Uniformities in the course of nature that don’t look like results of causation are, as I have shown, admissible as universal truths with a degree of belief proportioned to their generality. Those that are true of all things whatever, or at least are totally independent of the varieties of Kinds— namely the laws of number and extension, to which we may add the law of causation itself—are probably the only ones an exception to which is absolutely and permanently incredible. Accordingly, the word ‘impossible’ (or anyway ‘totally impossible’) seems usually to be confined to assertions regarded as contradictory to these laws or to others coming near to them in generality. Violations of other laws—of special laws of causation, for instance—are said by people who care about accuracy in speech to be ‘impossible in the circumstances of the case’ or ‘impossible except where there’s a cause that didn’t exist in the particular case’.[1] If a cautious person is faced with an assertion that doesn’t contradict any of these very general laws, he won’t go further than to call it ‘improbable’; and he won’t mean ‘improbable in the highest degree’ unless the time and place in which the fact is said to have occurred make it almost certain that the anomaly, if real, couldn’t have been overlooked by other observers. In any other case the judicious inquirer will avail himself of suspense of judgment, provided the testimony in favour of the anomaly presents, when well sifted, no suspicious circumstances.

The testimony hardly ever survives such a test in cases where the anomaly is not real. In the instances on record in which many witnesses of good reputation and scientific acquirements have testified to the truth of something that then turned out to be untrue there have almost always been details that would have made the testimony untrustworthy to a keen observer who had taken the trouble to sift the matter. There have generally been ways to explain the impression on the senses or minds of the alleged percipients, in terms of

•fallacious appearances, or

•some epidemic delusion propagated by the contagious influence of popular feeling, or

•some strong interest—religious zeal, party feeling, vanity, or at least the passion for the marvellous.

When nothing like that can account for the apparent strength of the testimony; and where the assertion

doesn’t contradict either •the universal laws that know no counteraction or anomaly or •the generalisations just below them in comprehensiveness,

but only

implies the existence of an unknown cause or an anomalous Kind, in circumstances where it is credible that hitherto unknown things may still come to light,

a cautious person will neither admit nor reject the testimony, but will wait for confirmation at other times and from other unconnected sources. That’s what the King of Siam should have done when the Dutch travellers told him about ice. But an ignorant person is as obstinate in his contemptuous incredulity as he is unreasonably credulous. Anything unlike his own narrow experience he disbelieves if it doesn’t answer to his needs or tastes; any nursery tale is swallowed implicitly by him if it does.

§4. I now come to a very serious misunderstanding of the principles of this subject that has been committed by some writers against Hume’s ‘Essay on Miracles’ and by Bishop Butler before them, in their anxiety to destroy what they saw as attack-weapon against the Christian religion. It has the effect of totally confusing the doctrine of the grounds of disbelief. The mistake consists in overlooking the distinction between . . . .the improbability that a mere guess is right and the improbability of an alleged fact being true. [The ellipsis in that sentence replaces ‘. . . (what may be called) improbability before the fact and improbability after it, or (since, as Venn remarks, the distinction of past and future is not the material circumstance) between. . . ’.]

Many events that are altogether improbable to us before they have happened or before we’re informed of their happening are perfectly credible when we are informed of them, because they aren’t contrary to any induction, even an approximate one. In the throw of a perfectly fair die, the chances are 5:1 against throwing 4; that is, 4 will be thrown on an average only once in six throws. But this is no reason against believing that ace was thrown on a given occasion if any credible witness asserts it. It’s true that 4 is thrown only once in six times, but if the die is thrown at all it must throw some number that is thrown only once in six times. The improbability (i.e. the unusualness) of any fact is no reason for disbelieving it if the situation makes it certain that either that or something equally improbable (i.e. equally unusual) did happen. Furthermore, even if the other five sides of the die are all 2s, still 4 would on the average come up once in every six throws, its coming up in a given throw would not in any way contradict experience. If we disbelieved all facts that had the chances against them beforehand, we would believe hardly anything. We are told that John Doe died yesterday; the moment before we were told this the chances against his having died on that day may have been 10,000:1; but since he was certain to die at some time, and when he died it had to happen on some particular day, experience gives us no basis for discrediting any testimony that may be produced to the event’s having occurred on 26.v.1872. The odds were against its happening on that day in particular, but only because they were against John Doe’s dying on day n for any value of n.

Yet George Campbell and others have offered as a complete answer to Hume’s doctrine that

things that are contrary to the uniform course of experience are incredible

the undisputed fact that we don’t disbelieve something that is in strict conformity with the uniform course of experience merely because the chances were against it; we don’t disbelieve an alleged fact merely because the combination of causes it depends on occurs only very infrequently. It’s obvious that whatever is shown by observation, or can be proved from laws of nature, to occur in a certain proportion (however small) of the whole number of possible cases is not contrary to experience; though we are right in disbelieving it if some other supposition regarding the matter in question takes us less far from the ordinary course of events. Yet on such grounds as this able writers have been led to the extraordinary conclusion that nothing supported by credible testimony ought ever to be disbelieved.

§5. I have considered two sorts of events that are commonly said to be improbable: one sort that are in no way extraordinary, but have an immense preponderance of chances against them and are therefore improbable until they are affirmed, but no longer; and another sort that are contrary to some recognised law of nature and are therefore incredible on any amount of testimony except such as would shake our belief in the law ·of causation· itself. But there’s also an intermediate class of events, consisting of what are commonly called ‘coincidences’—in other words, combinations of chances that present some special and unexpected regularity that makes them look like the results of law. An example would be, in a lottery with a thousand tickets, the numbers being drawn in the exact order 1, 2, 3, etc. We haven’t yet considered the principles of evidence that apply to this case—whether coincidences differ from ordinary events in the amount of testimony or other evidence necessary to make them credible.

It is certain that on every rational principle of expectation, a combination of this special sort may be expected quite as often as any other given series of a thousand numbers; that with perfectly fair dice, sixes will be thrown n times in succession (for any n) quite as often in a thousand or a million throws as any other succession of numbers fixed upon beforehand, and that no judicious player would give greater odds against the one series than against the other. [He means that the odds against throwing (for example) 6 6 6 6 are no greater than the odds against throwing 7 2 1 4 or 5 6 1 9 or. . . etc.] Yet there’s a general disposition to regard the one as much more improbable than the other, and as needing much stronger evidence to make it credible. This impression is so strong that it has led some thinkers to conclude that nature finds it harder to produce regular combinations than to produce irregular ones—i.e. that there’s some general tendency in things, some law, that prevents regular combinations from occurring as often as others. These thinkers include Jean D’Alembert, who in an essay on probabilities contends that regular combinations, though equally probable according to the •mathematical theory with any others, are •physically less probable. He appeals to common sense, i.e. to common impressions, saying that if a die thrown repeatedly in our presence gave sixes every time, before there had been ten throws (let alone thousands of millions) we would be absolutely sure that the die was loaded.

The common and natural impression is in favour of D’Alembert; the regular series would be thought much more unlikely than an irregular one. But this common impression is merely based on the fact that scarcely anyone remembers having ever seen one of these conspicuous coincidences. Why is that? It’s simply because no-one’s experience extends to anything like the number of trials within which that or any other given combination of events can be expected to happen. The chance of sixes on a single throw of two dice being 1/36, the chance of sixes ten times in succession is 1/36^10, which is to say that such an concurrence is only likely to happen once in 3,656,158,440,062,976 trials, a number that no dice-player’s experience comes up to a millionth part of. But if instead of sixes ten times some other given succession of ten throws had been fixed upon, it would have been exactly as unlikely that in any individual’s experience that particular succession had ever occurred; although this doesn’t seem equally improbable, because no-one would be likely to have remembered whether it had occurred or not, and because the comparison is tacitly made not between •sixes ten times and •any other particular series of ten throws, but between all regular successions and all irregular ones taken together.

D’Alembert is unquestionably right in saying that if the succession of sixes was actually thrown before our eyes we would ascribe it not to chance but to unfairness in the dice. But this arises from a totally different principle. What we should be asking is not

How probable was it that sixes would be thrown ten times in a row?

but rather

Given our knowledge that this did happen, how probable is it that the cause was C_1? C_2?. . . .etc.

The regular series is as likely as the irregular one to be brought about by chance, but it is much more likely than the irregular one to be produced by design or by some general cause operating through the structure of the dice. It is the nature of casual combinations to produce a repetition of the same outcome

as often as any other series of outcomes, and no oftener.

It is the nature of general causes to produce the same outcome

in the same circumstances, always.

Common sense and science alike dictate that other things being equal we should attribute the effect to •a cause which if real would be very likely to produce it rather than to •a cause that would be very unlikely to produce it. According to Laplace’s sixth theorem, which I demonstrated in chapter 18.5, the difference of probability arising from the greater efficacy of the constant cause, namely unfairness in the dice, would after a very few throws far outweigh any antecedent probability there could be against its existence.

D’Alembert should have put the question differently. He should have supposed that we had ourselves previously tested the dice, and knew by ample experience that they were fair. Another person then tries them in our absence, and assures us that he threw sixes ten times in succession. Is the assertion credible or not? Here the effect to be accounted for is not •the occurrence itself, but •the fact of the witness’s asserting it. This may arise either from its having really happened or from some other cause. What we have to estimate is the comparative probability of these two suppositions.

If this witness had reported having thrown some other series of ten numbers, assuring us that he took particular notice of the outcome of each throw, and if we regard him as generally truthful and careful, we would believe him. But the ten sixes are exactly as likely to have been really thrown as the ten other numbers, ·whatever they are·. So if the report (i) ‘I threw ten sixes in a row’ is less credible than (ii) ‘I threw the following ten-member sequence of numbers. . . ’ etc., the reason must be not that (i) is less likely than (ii) to be said truly but that it is more likely than (ii) to be said falsely.

One reason obviously presents itself why ‘coincidences’ are asserted falsely more often than ordinary combinations are. The coincidence arouses wonder. It gratifies the love of the marvellous. So the motives to lie—one of the most frequent of which is the desire to astonish—operate more strongly in favour of this kind of assertion than of the other kind. To that extent there’s clearly more reason to discredit an alleged coincidence than to discredit a statement which isn’t in itself more probable but which if it were made would not be thought remarkable. Sometimes, however, the presumption on this ground would be the other way. There are some witnesses who, the more extraordinary an occurrence might appear, would be the more anxious to check it with utmost care before venturing to believe it, and still more before asserting it to others.

§6. Laplace contends that a coincidence is not credible on the same amount of testimony as would justify us in believing an ordinary combination of events; and he bases this merely on the general ground that testimony is fallible, quite apart from any special chances of lying because of the nature of the assertion. To do justice to his argument I’ll need to illustrate it by the example chosen by himself.

If, says Laplace, there were 1000 ·numbered· tickets in a box, and one has been drawn out, then if an eye-witness says that the number drawn was 79 we find this credible even though the chances against it were 999:1. Its credibility is equal to the antecedent probability of the witness’s veracity. But if there were in the box 999 black balls and only one white, and the witness reports that the white ball was drawn, the case (according to Laplace) is very different—the credibility of his assertion is only a small fraction of what it was in the previous case. Laplace’s account of why occupies the next paragraph.

The nature of the case requires that the credibility these witnesses falls materially short of certainty. Let us suppose, then, that the credibility of the witness in the case we are considering is 9/10—that is, let us suppose that in every ten statements the witness makes, nine on an average are correct and one incorrect. Let us now suppose that there have been enough drawings to exhaust all the possible combinations, with our witness reporting on each outcome. In one case out of every ten in all these drawings he will have made a false announcement. But in the case of the thousand tickets, these false announcements will have been distributed impartially over all the numbers, and of the 999 cases in which 79 was not drawn, there will have been only one case in which it was announced. On the other hand, in the case of the thousand balls (the announcement being always either ‘black’ or ‘white’), if white wasn’t drawn and there was a false announcement, that false announcement must have been ‘white’; and since by the supposition there was a false announcement once in every ten times, ‘white’ will have been announced falsely in one-tenth of all the cases in which it wasn’t drawn, i.e. one-tenth of 999 cases out of every thousand. White, then, is drawn on an average exactly as often as ticket 79, but it is announced without having been really drawn 999 times as often as ticket 79; so the announcement requires much more testimony to make it credible.[2]

To make this argument valid we must suppose that the witness’s reports are average specimens of his general veracity and accuracy; or at least that they are neither more nor less so in the case of the black and white balls than in the case of the thousand tickets. But this assumption is not justified. A person is far less likely to go wrong if he has only one form of error to guard against than if he has 999 different errors to avoid. For instance, a messenger who might make a mistake once in ten times in reporting •the number drawn in a lottery might not err once in a thousand times if sent simply to observe •whether a ball was black or white. Laplace’s argument, therefore, is faulty even as applied to his own case. And that case is far from adequate as a stand-in for all cases of coincidence. Laplace has so contrived his example that though black answers to 999 distinct possibilities and white only to one, the witness has no bias that can make him prefer black to white. The witness didn’t know that there were 999 black balls in the box and only one white; or if he did, Laplace has taken care to make all the 999 cases so alike that any cause of falsehood or error operating in favour of any of them would almost certainly operate in the same way if there were only one. Alter this supposition, and the whole argument falls to the ground. Let the balls, for instance, be numbered, and let the white ball be 79. Considered in respect of their colour, there are only two things that the witness •can be interested in asserting, or •can have dreamed or hallucinated, or •has to choose from if he answers at random, namely black and white; but considered in respect of the numbers attached to them, there are a thousand; and if his interest or error happens to be connected with the numbers, though the only assertion he makes is about the colour, the case becomes precisely assimilated to that of the thousand tickets. Or instead of the balls suppose a lottery with 1000 tickets and only one prize, and that I hold ticket 79; because that’s all I am interested in, I ask the witness not ‘What number was drawn?’ but ‘Was ticket 79 drawn?’ There are now only two cases, as in Laplace’s example; but surely he wouldn’t say that if the witness answered ‘79’, the assertion would be enormously less credible than if he gave the same answer to the same question asked in the other way. . . .

Suppose a regiment of 1000 men, 999 Englishmen and one Frenchman, and that one of these has been killed and I don’t know which. I ask the question and the witness answers ‘It was the Frenchman’. This was as improbable a priori as the drawing of the white ball, and is also as striking a coincidence as that. But we would believe it as readily as if the answer had been ‘It was John Thompson’. The 999 Englishmen were all alike in the respect in which they differed from the Frenchman, but they weren’t indistinguishable in every other respect, as the 999 black balls were; and because they were all different there were as many chances of interest or error regarding them as if each man had been of a different nation; and if a lie was told or a mistake made, the misstatement was as likely to fall on any Jones or Thompson of the set as on the Frenchman.

D’Alembert’s example of a coincidence—sixes thrown on a pair of dice ten times in succession—belongs to this sort of case rather than to ones like Laplace’s. The coincidence here is much more remarkable, because of far rarer occurrence, than the drawing of the white ball. But though the improbability of its really occurring is greater, the greater probability of its being announced falsely can’t be established with the same evidentness. The announcement ‘black’ represented 999 cases, but the witness may not have known this, and even if he did, the 999 cases are so exactly alike that there’s really only one set of possible causes of mendacity corresponding to the whole. The announcement ‘sixes not drawn ten times,’ represents, and is known by the witness to represent, a great multitude of contingencies every one of which is unlike every other, so that there can be a different and a fresh set of causes of mendacity corresponding to each.

It appears to me therefore that Laplace’s doctrine is not strictly true of any coincidences, and is thoroughly false of most; and that to know whether a coincidence needs more evidence to make it credible than an ordinary event, we must refer in every instance to first principles, and estimate afresh what the probability is that the given testimony would have been given in that instance if the fact it asserts isn’t true.

With those remarks I close the discussion of the grounds of disbelief and, along with it, as much exposition of the logic of induction as space admits and I have it in my power to provide.

NOTES

[1] One writer. . . .defines ‘an impossibility’ as ‘that which there exists in the world no cause adequate to produce’. This definition doesn’t take in such impossibilities as that two and two should make five, that two straight lines should enclose a space, or that anything should begin to exist without a cause. I can’t think of any definition of ‘impossibility’ broad enough to include all its varieties, except the one I have given: An impossibility is something whose truth would conflict with a complete induction, i.e. with the most conclusive evidence we have of universal truth.

—As for the reputed impossibilities that rest purely on our ignorance of any cause that could produce the supposed effects: very few of them are certainly impossible or permanently incredible. The facts of travelling at 70 mph, painless surgical operations, and conversing by instantaneous signals between London and New York held a high place among such impossibilities not many years ago.

[2] But not 999 times as much testimony, as you might think. A complete analysis of the cases shows that (always assuming the veracity of the witness to be 9/10) in 10,000 drawings the drawing of ticket 79 will occur nine times and be announced incorrectly once; so the credibility of the announcement of ticket 79 is 9/10; while the drawing of a white ball will occur nine times, and be announced incorrectly 999 times. So the credibility of the announcement of white is 9/1008, which makes it only about 100 times more credible than the other, not 999 times.

Book IV: Operations Subsidiary to Induction

Glossary

Chapter 1. Observation and description

Chapter 2. Abstraction or the formation of conceptions

Chapter 3. Naming, as subsidiary to induction

Chapter 4. What is needed for a philosophical language. The principles of definition

Chapter 5. The natural history of the variations in the meaning of terms

Chapter 6. The principles of a philosophical language further considered

Chapter 7. Classification, as subsidiary to induction

Chapter 8. Classification by series

Glossary

agree: When Mill speaks of the respects in which two things ‘agree’, he means the respects in which they ‘are alike’. In this version, ‘agree’ will be allowed to stand sometimes, but it will often be replaced, for aesthetic reasons, by the language of ‘likeness’ and ‘resemblance’. Mill also speaks of a thing as ‘agreeing’ with a general conception, and he clearly means this to be a comparison also; in the present version he often says instead that the object ‘fits’ the conception.

art: Any practical activity that is governed by rules and (same thing?) requires skill. Mill’s ‘science or art’ might be illustrated by ‘physics or engineering’.

circumstance: Mill often uses this word as we do, but he also has a different use: the ‘circumstances’ of an event or state of affairs may be its features, details, facts about it. In those uses the present version usually replaces the word by something more familiar, but sometimes—e.g. in the opening paragraphs of chapter 5—there’s no way of doing that.

colligation: Collecting, gathering together, bundling.

connote: To say that word W connotes attribute A is to say that the meaning of W is such that it can’t apply to anything that doesn’t have A. For example, ‘man’ connotes humanity.

definite distinction: A distinction that can be captured in a definition. It ought also to be ‘definite’ in our ordinary sense—clear, sharp, unambiguous etc.—but the emphasis is on definability (unlike, say, the distinction between cows and horses).

denote: Refer to. The phrase ‘Raoul Wallenberg’ denotes a certain man; so does ‘that man standing near the window’ when said by someone as he points to a man standing by a window; and the abstract noun ‘humanity’ denotes a certain property.

disinterested: What this has meant for centuries, and still means when used by literate people, is ‘not self -interested’.

efficient cause: This is an Aristotelian technical term. The •formal cause of a coin is its design, the plan according to which it was made; its •material cause is the stuff it is made of; its •final cause is its purpose, namely to be used in commerce; and its •efficient cause is the action of the die in stamping the coin out of a metal sheet.

fact: Mill uses this word rather loosely, variously meaning ’state of affairs’, ‘alleged state of affairs’, and ‘proposition asserting the existence of a state of affairs’.

frame: To ‘frame’ a class or a conception is to create it, set it up, give it its conceptual structure.

oblate spheroid: A slightly flattened globe.

proximate: Next. The ‘proximate natural group’ relative to class C is the larger group, one step up, that contains C.

question-begging: Until fairly recently, to ‘beg the question’ was to offer a ‘proof’ of P from premises that include P. It now means ‘raise the question’. It seems that illiterate journalists (of whom there are many) encountered the phrase, liked it, guessed at its meaning, and saw no reason to check on the guess. For Mill’s lovely statement about this, see the indented passage here.

type: ‘The general form, structure or character distinguishing a particular group or class of things’ (OED).

vulgar: Ordinary not very intelligent and not very educated people. Here Mill says that it includes ‘all who have not accurate habits of thought’.

Chapter 1. Observation and description

§1. The inquiry that occupied us in Books II and III has led us to an apparently satisfactory solution of Logic’s chief problem, according to my conception of Logic. We have found that the mental process that Logic involves—the operation of ascertaining truths by means of evidence—is always a process of induction, even when appearances point to a different theory of it. And we have looked separately at the various types of induction, and obtained a clear view of the principles that it must obey if it is to lead to reliable results.

But there’s more to induction than the direct rules for performing it; something also has to be said about other mental operations that are either •presupposed in all induction or •instrumental in the more difficult and complicated inductive processes. The present Book will examines these subsidiary operations, starting with the ones that are indispensable preliminaries to all induction.

Induction is simply this:

extending to a class of cases something that has been •observed to be true in some members of the class.

So •observation is the first subsidiary operation to be looked at. But we shan’t be laying down rules for observing well; those are within the province not of logic but of intellectual education. Where observation connects with logic is in the evaluation of evidence. We shan’t be asking •How are we to observe? or •What should we observe? but rather •When is observation to be relied on? What is needed for a report of a supposed observation to be safely accepted as true?

§2. The answer to this question is very simple, at least in its first aspect [Mill’s phrase]. The only condition is that what is supposed to have been observed shall really have been observed, i.e. that it was an observation and not an inference. In almost every act of our perceiving faculties, observation and inference are intimately blended. What we’re said to observe is usually a compound result, sometimes one-tenth observation and nine-tenths inference.

. . . .I affirm that I saw my brother at 9 am this morning. Isn’t this something that I know through the direct testimony of my senses? The ordinary answer would be a confident ‘Yes!’, but in fact that answer is wrong. All I saw was a certain coloured surface; or rather I had visual sensations of a kind that is usually produced by a coloured surface; and on the basis of my previous experience of such sensations I concluded that I saw my brother. I could have had sensations just like those when my brother wasn’t there—perhaps seeing some other person very like him. I might have been asleep and have dreamed that I saw him; or in a state of nervous disorder that brought his image before me in a waking hallucination. In all these ways many people have been led to believe that they saw familiar friends or relatives who weren’t actually these. If any of these suppositions had been true, the statement that I saw my brother would have been false, though the direct perception—the visual sensations—would have been real. The only trouble would have been that my inference was ill-grounded; I would have assigned a wrong cause for those sensations.

Countless instances might be given, and analysed in the same manner, of what are vulgarly [see Glossary] called ‘errors of sense’. None of them are strictly errors of sense—they are erroneous inferences from sense. When I look at a candle through a multiplying glass, I seem to see a dozen candles instead of one; and if the set-up were skillfully disguised, I might suppose that there were really that number—that would be what is called an ‘optical deception’. In the kaleidoscope there really is that deception; when I look through the instrument, the appearance presented to me is not •a casual arrangement of coloured fragments but rather •a single combination of coloured bits repeated several times in a symmetrical arrangement around a point. The delusion is caused by my having the same sensations that I would have had if such a symmetrical combination had really been presented to me. If I cross two of my fingers and bring a marble (say) into contact with both at points that aren’t usually touched simultaneously by one object, I can hardly help believing—if my eyes are shut—that there are two marbles rather than one. But what gets deceived is not my sense of touch (in this case) or of sight (in the other), but rather my judgment, and that’s true even if the deception is only momentary. All I get from my senses are the sensations, and they are genuine. I have been accustomed to having sensations like those whenever a certain arrangement of outer objects is present to my sense-organs, and not at any other time; so I’ve formed the habit of instantly inferring the existence of that state of outer things whenever I experience such sensations. This habit has become so powerful that the •inference, performed with the speed and certainty of an instinct, is taken to be an intuitive •perception. When its conclusion is correct, I’m not aware that it ever needed proof; and even when I know that it’s incorrect, it’s quite difficult for me to abstain from conducting the inference that leads to it. To become aware that the judgment in question is made not by instinct but by an acquired habit, I have to think about •the slow process through which I learned to judge by the eye concerning many things that I now appear to perceive directly by sight; and about •the reverse process that persons learning to draw go through—the difficult and laborious task of shedding their acquired perceptions and learning afresh to see things as they appear to the eye.

From these examples (and I could give ever so many more) we can see that the individual facts from which we derive our inductive generalisations are hardly ever obtained by observation alone. Observation extends only to the sensations by which we recognise objects; but the propositions that we make use of in science or in common life mostly relate to the objects themselves. In every act of ‘observation’ there’s at least one inference—from the sensations to the presence of the object, i.e. from the marks or pointers to the entire phenomenon. That leads to a seeming paradox:

A general proposition derived from particulars is often more certainly true than any one of the particular propositions that it was inferred from by an act of induction.

Each of those particular propositions involved an inference from •the impression on the senses to •the fact that caused this impression; and this inference may have been wrong in any one of the instances, but can hardly have been wrong in all of them, provided there were enough of them to eliminate chance. It follows that the general proposition may deserve more complete reliance than could safely be given to any one of the inductive premises.

So all there is to the logic of observation is a correct discrimination between what is really perceived and what is inferred from that. The inferential part of this falls under the rules of induction which I have already discussed and needn’t go into again. What I do have to confront is this question: ‘When all the inference is taken away what remains?’ Well, there’s this:

the mind’s own feelings or states of consciousness— its outward feelings or sensations and its inward feelings—its thoughts, emotions, and volitions.

Another question arises: ‘Is that all that remains? Is that the sole basis for our inferences to other things? Or is our mind capable of directly perceiving or grasping something other than the states of its own consciousness?’ That is a problem of metaphysics that won’t be discussed here. But after setting aside all questions on which metaphysicians differ, it remains true that for most purposes what we need in practice is to distinguish •sensations or other feelings (ours or those of others) from •inferences drawn from them. And that seems to be all that needs to be said in the present work regarding the theory of observation.

§3. If in the simplest cases of what we ordinarily count as ‘observation’ there’s a large part that is not observation but something else, so also in the simplest description of an observation there must always be much more asserted than is contained in the perception itself. [The last 10 words are verbatim from Mill; they are imperfect, but what he is deriving at becomes clear right away:] We can’t describe a fact [see Glossary] without implying more than the fact. The perception is only of one individual thing, but describing it involves affirming a connection between it and every other thing that is either denoted or connoted by any of the terms used [see Glossary for: connote and denote]. There couldn’t be a more elementary example, than this: I have a visual sensation which I try to describe by saying ‘I see something white’. In saying this I don’t merely •affirm my sensation—I also •classify it. I assert that the thing I see resembles and all the things that I and others are accustomed to calling ‘white’. I say that it resembles them in the respect in which they all resemble one another—the respect that is the basis for calling them ‘white’. This isn’t just one way of describing an observation; it’s the only way. If I want to record my observation (for my own future use or to inform others) I must assert a resemblance between the fact which I have observed and something else. It is inherent in a description to be the statement of a resemblance, or resemblances.

So we see that. . . .we can’t speak of an observation—can’t bring it under language—without declaring more than that one observation, which we do by assimilating it to other phenomena already observed and classified. Whewell regards this ·going-from-one-to-many· process as characteristic of induction. But ·he is wrong·: this identification of an object—this recognition of it as having certain known characteristics—has never been muddled with induction. It’s a perception of resemblances, obtained by comparison; it precedes all induction, and supplies it with its materials.

These resemblances aren’t always grasped directly, by merely comparing the object observed with some other present object or with our memory of an object that isn’t present. They are often learned. . . .deductively. In describing some new kind of animal I say that it is ten feet long from the forehead to the tip of the tail. I didn’t learn this from my unassisted eye. I had a two-foot ruler which I applied to the object and measured it; and rather than being wholly manual the measuring operation was partly mathematical, involving the two propositions Five times two is ten and Things that are equal to the same thing are equal to one another. So the fact that the animal is ten feet long is not an immediate perception, but a conclusion from reasoning; the observation of the object provides only the other premises of the inference. But we call this an observation, or a description of the animal, not an induction concerning it.

Now for a very complex example: I affirm that the earth is globular. The assertion isn’t based on direct perception; indeed, we can’t directly perceive the shape of the earth (though the assertion wouldn’t be true unless there were conceivable circumstances in which its truth could be perceived directly). That the earth is globular in shape is inferred from certain marks, such as:

•its shadow thrown upon the moon is circular;

•on the sea or any large plain our horizon is always a circle;

each of which is incompatible with any shape except that of a globe. I then go further, and say that the earth is an oblate spheroid [see Glossary], which is one kind of globe [and he gives a very unclear statement of the evidence for this]. But each of these propositions—The earth is globular and The earth is an oblate spheroid—asserts an individual fact which could be perceived by the senses if we had the required sense-organs and the needed viewpoint; so each could properly enough be called a ‘description’ of the earth’s shape, even though it has been inferred rather than seen. But it wouldn’t be proper to call either of these assertions an ‘induction’ from facts about the earth. They aren’t general propositions inferred from particular facts, but particular facts inferred from general propositions. They are conclusions deduced from premises originating in induction, ·but don’t think ‘So this is all a matter of induction after all, with observations of the earth as its basis’·: some of those premises weren’t obtained by observation of the earth and had no special reference to it.

Why should the truth about the shape of the earth’s orbit be an induction if the truth about its own shape is not? The two cases differ in this:

•The shape of the earth was established by reasoning from facts that were signs of ellipticity;

•The shape of the earth’s orbit was established by first guessing that it was an ellipse and then finding that empirical observations confirmed that hypothesis.

According to Whewell, however, this process of guessing and verifying our guesses is induction, and is indeed the whole of induction: no other exposition (he thinks) can be given of that logical operation. Well, the whole Book III of the present work has, I hope, shown that another account can be given; and I have tried in chapter 2 of that Book to show that the process by which the ellipticity of the planetary orbits was learned is not induction at all. Now, however, I can go deeper into the heart of the matter and show not merely what that process is not but what it is.

§4. I remarked in III.2 that the proposition ‘The earth moves in an ellipse’ can be taken in either of two ways:

•as a description which serves for the colligation [see Glossary] of actual observations—i.e. merely says that the observed positions of the earth can be correctly represented by points along an imaginary ellipse; or

•as an induction which says that positions of the earth that haven’t yet been directly observed would be found to correspond to the remaining points on the same ellipse.

The induction is one thing and the description another; but we’re in a much better position to conduct the induction if we already have the description. That is because the description—like all descriptions—implies a resemblance between the thing described and something else; in pointing out a resemblance among •the observed places of the earth, it points out something in which •all its places ·may· agree [see Glossary]. . . . So we have, by the same process that gave us the description, obtained what we need for an induction by the Method of Agreement ·that I introduced in III:8.1·. Considering the observed places of the earth as effects, and its motion as the cause that produces them, we find that those effects are all in an ellipse and conclude that the remaining effects—the places that haven’t yet been observed—are all in that ellipse and that the law of the earth’s motion is motion in an ellipse.

Thus the colligation of facts by means of hypotheses. . . .takes its proper place among operations subsidiary to induction. All induction presupposes that we have already compared the required number of individual instances and discovered the respects in which they agree, and this preliminary operation is the colligation of facts. Kepler tried in vain to connect the observed places of a planet by various hypotheses of •circular motion, and then at last he tried the hypotheses of an •ellipse and found that it squared with the phenomena. What was really going on here was an attempt— at first unsuccessful and then successful—to discover the respect in which each planet’s observed positions agreed [see Glossary] with one another. Then he connected another set of observed facts, the times it took the different planets to complete their orbits, by the proposition that the squares of the times are proportional to the cubes of the distances, thus ascertaining the property in which the periodic times of all the different planets. [Amplifying Mill’s short-hand a little: for each planet the relevant ‘distance’ is half the length of the longest straight line through that planet’s ellipse.]

All that is true and relevant in Whewell’s doctrine of ‘conceptions’ can be fully expressed by the more familiar term ‘hypotheses’; and his ‘colligation of facts by means of appropriate conceptions’ is just the ordinary process of comparison that I have been describing. So I could have confined myself to those better understood expressions ·and left Whewell out of the discussion·. I’d have been glad to do that, staying with my policy in the present work of avoiding ideological discussions and treating •the mechanism of our thoughts to be irrelevant to •the principles and rules by which the trustworthiness of the results of thinking is to be estimated. But such ideological considerations are the sole basis for a theory of induction in a work—·Whewell’s Novum Organum Renovatum·—which makes very large claims and has indeed much real merit; so those who come after him have to claim for themselves and their doctrines whatever position may properly belong to them on the same metaphysical ground. That is the aim of the next chapter.

Chapter 2. Abstraction or the formation of conceptions

§1. The metaphysical inquiry into the nature and composition of what have been called ‘abstract Ideas’, i.e. the notions that correspond in the mind to classes and to general names, belongs not to logic but to a different science; and my purpose here doesn’t require me to enter upon it. My only concern is with something everyone accepts, namely that such general notions or conceptions do exist. The mind can conceive a multitude of individual things as one assemblage or class; and general names do really suggest to us certain ideas or mental representations, otherwise we couldn’t use the names with consciousness of a meaning. Whether the idea called up by a general name is

•composed of the various respects in which all the individuals denoted by the name are alike, and of no others (Locke, Brown, and the conceptualists), or

•the idea of some one of those individuals, clothed in its individual features and accompanied by the knowledge that those features aren’t properties of the class (Berkeley, Bailey, and the modern nominalists), or

•the idea of a miscellaneous assemblage of individuals belonging to the class (James Mill), or

•any one or any other of all these, according to the accidental circumstances of the case,

it is certain that some idea or mental conception is suggested by a general name whenever we either hear it or use it ourselves with consciousness of a meaning. And this general idea represents in our minds the whole class of things to which the name is applied. Whenever we think or reason concerning the class, we do so by means of this idea. And our mind’s ability to attend to one part of what is present to it and neglect the rest enables us to keep our reasonings and conclusions regarding the class unaffected by anything in the idea that isn’t really, or at least that we don’t really believe to be, common to the whole class.

So there are such things as general conceptions, or conceptions through which we can think generally; and when we bring a set of phenomena into a class—i.e. compare them with one another to discover what they agree in—some general conception is implied in this mental operation. Given that such a comparison is a necessary preliminary to induction, it is most true that induction couldn’t happen without general conceptions.

§2. But it doesn’t follow that these general conceptions must have existed in the mind before the comparison. It isn’t a law of our intellect that

when we compare things and notice their similarities, we’re merely recognising instances in the outer world of something that we already had in our minds.

The conception originally found its way to us as a result of such a comparison; we acquired it by abstraction (a metaphysical term) from individual things. These may be things that we perceived or thought of on earlier occasions, but they may be the things we are perceiving or thinking of on the very occasion. When Kepler compared the observed positions of Mars and found that they agreed in being points on an ellipse, he was applying a general conception that was already in his mind because he had derived it from his previous experience. But this is by no means always the case. When we compare some objects and find them to be alike in being white, or compare the various species of cud-chewing animals and find them to be alike in being cloven-footed, we have just as much a general conception in our minds as Kepler had in his: we have the conception of a white thing or a cloven-footed animal. But no-one thinks that we have to bring these conceptions with us and apply them to the facts from the outside; because in these simple cases everyone sees that •the very act of comparison that leads us to connect the facts by means of the conception may be •the source from which we derive the conception. If we had never seen any white object (or any cloven-footed animal) we would at the same time and by the same mental act •acquire the idea and •use it for the colligation of the observed phenomena. Kepler, on the other hand, really had to bring the idea with him and apply it to the facts from the outside; he couldn’t evolve it out of the facts—couldn’t have acquired it by comparing the planet’s positions. But this inability was a mere accident, ·a contingent feature of this particular case·: Kepler could have been acquired the idea of an ellipse from the paths of the planets if those paths hadn’t happened to be invisible; if a planet had left a visible track, and he had been placed so that he could see it at the proper angle, he could have abstracted his original idea of an ellipse from the planet’s orbit. Indeed, any conception which can be used as an instrument for connecting a set of facts could have been originally developed out of those very facts. The conception is a conception of something; and what it’s a conception of is really in the facts and could—in some supposable circumstances, or by some supposable extension of our actual faculties—have been detected in them. And not only is this always in itself possible, but it actually happens in most cases where it’s difficult to obtain the right conception. If no new conception is required—if the job can be done by a conception that is already familiar to mankind—almost anyone might happen to be the first to think of the right one, at least in the case of a set of phenomena that the whole scientific world are trying to connect. The honour, in Kepler’s case, was that of the accurate, patient, laborious calculations by which he compared •the results that followed from his different guesses with •the observations of Tycho Brahe. Guessing an ellipse wasn’t a great achievement: it would have been guessed long before his time if there hadn’t been an obstinate a priori prejudice that the heavenly bodies must move in a circle or some combination of circles.

The really hard cases are the ones where the conception destined to create light and order out of darkness and confusion has to be looked for among the very ·dark and confused· phenomena that it then serves to arrange. Why, according to Whewell, did the ancients fail to discover the laws of mechanics, i.e. the laws of equilibrium and of the communication of motion? Because they didn’t have clear ideas or conceptions of •pressure and resistance, •momentum, and •uniform and accelerating force. Where could they have acquired these ideas from if not from the very facts of equilibrium and motion? The late development of several of the physical sciences—e.g. optics, electricity, magnetism—and the higher generalisations of chemistry Whewell ascribes to the fact that mankind didn’t yet have the idea of polarity, i.e. of opposite properties in opposite directions. But what was there to suggest such an idea, until the separate pursuit of several of these sciences revealed that some of the facts of each of them did present the curious phenomenon of opposite properties in opposite directions? This was obvious on the surface in only in two cases, those of the magnet and of electrified bodies; and there the conception was cluttered ·and somewhat hidden· by the fact that the opposition of properties seemed to be inherent in material poles, fixed points in the body itself. The first comparison and abstraction led only to this conception of poles; and if anything corresponding to that conception had existed in the phenomena of chemistry or optics, the great difficulty would have been extremely small. What created the difficulty was the fact that

the polarities in chemistry and optics

were distinct species from

the polarities in electricity and magnetism,

though of the same genus. To bring all these domains under a single theory, it was necessary to compare a polarity without poles (e.g. the polarisation of light) with the (apparent) poles that we see in the magnet; and to recognise that these different polarities have something in common, namely the character that is expressed by the phrase ‘opposite properties in opposite directions’. It was from the result of such a comparison that scientists formed this new general conception. To get to •that from •the first confused feeling of an analogy between some of the phenomena of light and those of electricity and magnetism took a long time and much work and more or less clever suggestions by many superior minds.

So the conceptions we use for collecting and organising facts don’t grow from within but are impressed upon the mind from the outside; they are never obtained except through comparison and abstraction, and in most cases— including the most important ones—they are evolved by abstraction from the very phenomena that it is their role to gather together. I don’t deny •that it’s often very difficult to do this abstraction well, or •that the success of many inductive operations depends mainly on how well the abstraction was done. Bacon was quite justified in designating ‘general conceptions wrongly formed’ as one of the chief obstacles to good induction. [Mill also quotes Bacon’s Latin, notiones temere a rebus abstractæ, which more strictly means ‘notions abstracted carelessly (or casually) from things’.]. . . .

§3. As I try to show show what the difficulty in this matter really is, and how it is overcome, please bear this in mind: when I’m discussing a different school of philosophy I’m willing to adopt their language, so that I’ll speak of

‘connecting facts through the instrumentality of a conception’,

this technical terminology means neither more nor less than what is commonly called

comparing the facts with one another and determining what they agree in.

And the technical expression doesn’t even have the advantage of being metaphysically correct. The facts aren’t ‘connected’ except in a metaphorical sense of the word. The ideas of the facts may become connected, i.e. we may be led to think of them together; but this could be the result of any casual association. What really happens is more philosophically expressed, I think, by the common word ‘comparison’ than by the phrase ‘to connect’. . . . We acquire the general conception by •comparing particular phenomena, and then, once we have it, we apply it to other phenomena by •comparison. . . . We get the conception of an animal (for instance) by comparing different animals, and when we then see a creature resembling an animal we compare it with our general conception of an animal; and if it fits that general conception we include it in the class ·of animals·. The conception becomes the type [see Glossary] of comparison. And when you consider what it is to compare things, you’ll see that when indefinitely many objects are being compared—and even when only three are being compared— the comparison can’t be done without a type of some sort. When we have to arrange and classify very many objects according to their agreements and differences, we don’t make a confused attempt to compare each of them with all the others. We know that the mind can’t easily attend to more than two things at a time; so we fix on one of the objects (either randomly or for a reason) and, taking this as our standard ·or type·, compare it with one object after another. If we find a second object that agrees remarkably with the first, leading us to class them together, the question instantly arises: In what particular respect do they agree? Answering that is already a first stage of abstraction, giving rise to a general conception. Having gone that far, we now attend to a third object, and ask: Does the third object agree with the first in the same respect in which the second did? That is, does it fit the general conception that has been obtained by abstraction from the first and second? This shows the tendency of general conceptions, once we have them, to serve as types in place of whatever individual objects we previously used in that way in our comparisons. If we find that not many objects fit this first general conception, we drop it and start again. . . . Sometimes we find that a conception will serve if we leave out some of its details; and by this higher effort of abstraction we obtain a still more general conception. I have given an example of this: the scientific world’s ascent from the conception of •poles to the general conception of •opposite properties in opposite directions. . . .

These brief remarks contain, I believe, all that is solid in the theory that the conception that the mind arranges and unifies phenomena by must be provided by the mind itself, and that we find the right conception by trying first one and then another until we hit the mark. The conception isn’t provided by the mind until it has been provided for the mind; and the facts that supply it are mostly (not exclusively) the very facts we are trying to arrange by it. It’s true, though, that in trying to arrange the facts we never advance three steps without forming •a general conception, more or less clear and precise; and that •this becomes the ·type or· standard in terms of which we then compare the rest of the facts. If we aren’t satisfied with the agreements that we find among the phenomena by comparing them with this type, or with some more general conception that we can form from this type by a further stage of abstraction, we change our path and look out for other agreements. We re-start the comparison from a different starting-point, and so generate a different set of general conceptions. This is the tentative process that Whewell speaks of, and it’s not surprising that it has suggested the theory that the conception is supplied by the mind itself. Whenever the mind puts a conception to work in comparing two things, it’s either •a conception that it already has from its previous experience or •one that was supplied to it in the first stage of this very comparison; so that in the later parts of the process the conception presents itself as something compared with the phenomena, not evolved from them.

§4. If this is a correct account of the ‘instrumentality of general conceptions’ in the comparisons that necessarily precede induction, I can now translate into plain language what Whewell means by saying that if a conception is to be useful in induction it must be ‘clear’ and ‘appropriate’.

If the conception corresponds to a real likeness among the phenomena—if the comparisons we have made among of a set of objects has led us to class them according to real resemblances and differences—the conception that does this is bound to be ‘appropriate’ for some purpose or other. Appropriateness is a relative matter: it depends on what we are trying to do. As soon as our comparisons show us something that can be predicated of a number of objects, we have a basis on which an inductive process could be founded. [But it’s a further question, Mill goes on to say, whether the induction will be worth doing. He contrasts classifying animals by colour with classifying them by skeletal structure. He continues:] Agreements and differences in respect of skeletal structure are not only more important in themselves ·than colour is·, but they are marks of agreements and differences in many other important features of animals. If the latter features are what we want to study, the conceptions generated by skeletal-structure comparisons are far more ‘appropriate’ than those generated by comparisons in respect of colour. This is all that can be meant by the ‘appropriateness’ of a conception.

When Whewell says that philosophers of this or that school missed discovering the real law of a phenomenon because the conception they brought to bear on it was ‘inappropriate’, he can only mean that in comparing various instances of the phenomenon so as to discover what those instances agreed in they missed the important points of agreement, and (at best) fastened onto ones that were comparatively trifling. . . .

Aristotle distinguished two sorts of motion, which he called ‘natural’ and ‘violent’ respectively.

•Natural motions apparently take place spontaneously— bodies fall to the ground, flame ascends, bubbles of air rise in water, etc.

•Violent motions never occur without external incitement, and tend spontaneously to cease.

In comparing the ‘natural’ motions with one another, Aristotle thought that they agreed in one respect, namely, that the body that moved (or seemed to move) spontaneously was moving toward its own place, meaning the place it originally came from or the place where a great quantity of matter similar to itself was assembled. In ‘violent’ motions on the other hand, e.g. when bodies are thrown up in the air, they are moving from their own place. This conception of a body moving toward its own place can fairly be called ‘inappropriate’, for three reasons. It does express a feature that really is found in some of the most familiar instances of apparently spontaneous motion, but

(1) in many cases of such motion that feature is absent, e.g. the motion of the earth and planets.

(2) In many cases where the feature is present, the motion turns out not to be spontaneous. For example, when air rises in water it doesn’t rise by its own nature but is pushed up by the superior weight of the water pressing on it.

(3) The spontaneous motion often occurs in the opposite direction to what the theory regards as the body’s ‘own place’—e.g. when a fog rises from a lake, or when water dries up.

So the agreement that Aristotle selected as his principle of classification didn’t cover all cases of the phenomenon he wanted to study, spontaneous motion; and did cover cases where the motion is not spontaneous. The conception, in short, was ‘inappropriate’. I would add that in this case no conception would be appropriate: there’s no agreement running through all the cases of spontaneous or apparently spontaneous motion and no others; they can’t be brought under one law; this is a case of plurality of causes.

§5. So much for the first of Whewell’s conditions, that conceptions must be ‘appropriate’. The second is that they must be ‘clear’: let us consider what this implies. A conception that doesn’t correspond to a real agreement is irrelevant to our purposes; so let us suppose that among the phenomena that we are trying to connect by means of conception C •there really is an agreement and •C is a conception of it. For C to be clear, then, all that is needed is for us to know exactly what the agreement consists in—for us to have carefully observed and accurately remembered it. We are said not to have a ‘clear’ conception of the resemblance among a set of objects when we have only a general feeling that they resemble, without having analysed their resemblance—i.e. perceived what details it consists in—and fixed in our memory an exact recollection of those details. This lack of clearness, which we could call this vagueness, in the general conception may come from (a) our having no accurate knowledge of the objects themselves or merely from (b) our not having carefully compared them. Thus a person may have no clear idea of a ship because (a) he has never seen one or has only a faint and sketchy memory of the ones he has seen, or because (b) he has perfect knowledge and memory of many ships of various kinds, frigates included, but has no clear idea of a frigate because he hasn’t taken in and remembered what the differences of detail are between frigates and other ships.

Still, you can have a clear idea without knowing all the common properties of the things which we use it to class together. That knowledge would involve having a conception of the class that was complete as well as clear. All that is needed ·for clarity· is •that we never class things together without knowing exactly why we do so—without having settled exactly what agreements we’re going to include in our conception, and •that after thus fixing our conception we don’t vary from it by including in the class anything that lacks those common properties or excluding from it anything that has them. A ‘clear’ conception is a determinate conception—one that doesn’t fluctuate, that isn’t one thing today and another tomorrow, but remains fixed and invariable except when we consciously add to it or alter it because of something we have learned. . . .

What are mainly needed for clear conceptions, therefore, are •habits of attentive observation, •extensive experience, and •a memory that takes in and retains an exact image of what is observed. The more someone has of those virtues in relation to a particular class of phenomena, the clearer his conceptions of them will be. He must also never use general names without a precise connotation, but that will naturally result from those other endowments.

As the clarity of our conceptions mainly depends on how careful and accurate our observing and comparing faculties are, their appropriateness—or rather our chance of hitting on the appropriate conception in any given case—mainly depends on how active those same faculties are. If someone has, by habit based on sufficient natural aptitude, become skilled in accurately observing and comparing phenomena, he will perceive so many more agreements, and will perceive them so much faster than other people, that he has a much greater chance of perceiving, in any instance, the agreement that the important consequences depend on.

§6. It is so important that the topic of this chapter should be rightly understood that I think I should restate in a somewhat different way the results I have arrived at.

We can’t discover general truths, i.e. truths about classes, unless we have formed the classes in such a way that general truths can be true of them. Forming any class involves conceiving it as a class, i.e. conceiving certain features as being those that characterize the class and distinguish its members from all other things. When we know exactly what these features are, we have a clear idea (or conception) of the class and of the meaning of the general term that names it. The main requirement for having this clear idea is that the class really is a class; that it corresponds to a real distinction; that the things it includes really do agree with one another in certain respects and differ (in those same respects) from all other things. A person without clear ideas is one •who habitually classes together under the same general names things that have no common properties, or none that aren’t possessed also by other things; or •who, if the usage of other people prevents him from actually misclassing things, can’t state to himself the common properties on the basis of which he classes them rightly.

But there’s more to a good classification than merely picking out a real class framed [see Glossary] by a legitimate mental process. Some ways of classifying things are more useful to us than others, and our classifications aren’t well made unless the things they bring together don’t just

•agree with each other in something that distinguishes them from all other things, but also

•agree with each other and differ from other things in the very respects that are of primary importance for the purpose we have in view.

In other words, even our clear conceptions are not appropriate for our purposes unless the properties we build into them are ones that will help us toward our goal—i.e. that go deepest into the nature of the things, if we’re trying to understand that, or that are most closely connected with the particular property we are trying to investigate. [Twice in this paragraph Mill specifies that all this covers not only ‘speculative’ goals and pursuits but also ’practical’ ones, distinguishing the sciences on the one hand from the likes of morals and politics on the other.]

So we can’t frame good general conceptions beforehand. ‘This conception that I have obtained, is it the one I want?’: you can’t answer that until you have done the work you want it for, i.e. until you completely understand the general character of the phenomena, or the conditions of the particular property that you are studying. General conceptions formed without this thorough knowledge are Bacon’s notiones temere a rebus abstractæ, but we must be continually making up such premature conceptions in our progress to something better. They aren’t a drag on the progress of knowledge unless they are permanently accepted. When it has become our habit to group things in wrong classes—in groups that aren’t really classes, having no distinctive points of agreement (absence of clear ideas), or aren’t classes of which anything important to our purpose can be predicated (absence of appropriate ideas)—and when, believing that these badly made classes are sanctioned by nature, we •refuse to exchange them for others and •can’t or won’t make up our general conceptions from different materials, then all the evils that Bacon ascribes to his notiones temere abstractæ really occur. This is what the ancients did in physics, and what the world in general does in morals and politics even today.

I therefore don’t think it is right to say that we have to obtain appropriate conceptions before we start generalising. All through the process of comparing phenomena for the purpose of generalisation, the mind is trying to make up a conception; but the conception it’s trying to make up is that of the really important respect in which the phenomena agree. As we learn more about the phenomena themselves, and the conditions their important properties depend on, our views about this naturally alter; and thus we advance from a less to a more ‘appropriate’ general conception as our investigations progresses.

But don’t forget that the really important agreement can’t always be discovered by mere comparison of the very phenomena in question, without help from a conception acquired elsewhere. We saw this with the planetary orbits.

The search for the agreement of a set of phenomena is in truth very similar to the search for a lost or hidden object. At first we get into a commanding position and look around from there. If we don’t see the object, we ask ourselves where it might be hidden, so as to look for it there; and so on, until we imagine the place where it really is. In this procedure we need a previous conception, or knowledge, of those different places. This illustrates the philosophical operation in which we first try to find the lost object or recognise the common attribute, without conjecturally invoking the aid of any previously acquired conception, i.e. of any hypothesis. Having failed in this, we call on our imagination for some hypothesis of a possible place, or a possible respect of resemblance, and then look to see whether the facts fit the conjecture.

For such cases something more is required than a mind accustomed to accurate observation and comparison. It must be a mind already stored with general conceptions that have some relation to the subject of the particular inquiry. And much will also depend on the natural strength and acquired culture of what has been termed the scientific imagination— i.e. the ability to mentally arrange known elements into new combinations that haven’t yet been observed in nature but don’t conflict with any known laws.

But the variety of intellectual habits, the purposes they serve, and the ways they can be developed, are themes belonging to the art of education, a subject far wider than logic, and one that this treatise doesn’t claim to discuss. So the present chapter can properly close here.

Chapter 3. Naming, as subsidiary to induction

§1. I don’t need here to dwell on the importance of language for expressing sympathy and giving information. And I need only a passing mention of language’s power to form and rivet associations among our ideas, this being the ultimate source of their functions as intellectual instruments. [Mill’s ‘passing mention’ of this is a long quotation from that ‘able thinker’ Alexander Bain, writing about the services language does for us: anchoring and storing things that might otherwise slip out of out minds, prodding us to notice likeness that we might have overlooked, and so on. Bain concludes:] ‘The number of general names in a language, and the degree of their generality, provides a test of the knowledge of the era and of the intellectual insight that is the birthright of anyone born into it.’

What I have to discuss, however, is not the functions of names, considered generally, but only the topic of how and how much they are directly instrumental in the investigation of truth, i.e. in the process of induction.

§2. Observation and Abstraction, which have had a chapter each, are indispensable to induction; it can’t be done without them. Some thinkers have held that the same is true of Naming. . . . In their view, names or at least some kind of artificial signs are necessary for reasoning: there could be no inference, and thus no induction, without them. But if I was right in my account of reasoning in Book II, then this opinion must be regarded as an exaggeration, though of an important truth. If reasoning is from particulars to particulars, and if it consists in recognising one fact as a mark of another or as a mark of a mark of another, nothing is needed to make reasoning possible except senses and association; •senses to perceive that two facts are conjoined, and •association as the law by which one of those two facts raises up the idea of the other. There is evidently no need of language for these mental phenomena, or for the belief or expectation that follows them;. . . .and this inference of one particular fact from another is a case of induction. The lower animals are capable of this sort of induction; it’s the sort that uncultivated minds nearly always conduct; and we all do so in the cases where familiar experience forces our conclusions upon us without any active inquiry on our part, and cases where the belief or expectation follows the evidence with the speed and certainty of an instinct.[1]

§3. Although inductive inference without the use of signs is possible, it couldn’t get far beyond the very simple cases I have just described—cases that almost certainly form the limit of the reasonings of animals that don’t have conventional language. Without language or some equivalent of it, there could be only as much reasoning from experience as can take place without the aid of general propositions. Strictly speaking, we can reason from past experience to a fresh individual case without going through a general proposition, but without general propositions we would seldom remember •what past experience we have had or •what conclusions it warrants. [If what Mill wrote next concerns general propositions it is utterly unclear. After it he writes clearly about language:] The experience by which we’re to guide our judgments may be other people’s experience, which we can’t know much about except through language; and when the experience is our own, it is generally experience long past, so that in most cases little of it would be retained in the memory unless it were recorded with artificial signs. (The other cases are ones involving our intenser sensations or emotions, or things that we think about daily and hourly.) I hardly need to add that when an inductive inference requires (as most of them do) comparisons among several observations or experiments in varying circumstances, we can’t move a step without the artificial memory that words provide. If we often see A and B in immediate and obvious conjunction, we don’t need language to be led to expect B whenever we see A; but

•to discover their conjunction when it isn’t obvious, or

•to discover whether it is really constant or only casual, and

•whether there’s reason to expect it under any given change of circumstances

is far too complex a process for us to perform without some contrivance to give accuracy to our memory of our own mental operations. Now, language is such a contrivance. When that instrument is called to our aid, the difficulty shrinks to that of accurately remembering the meanings of words. . . .

[This section ends with a repetition in slightly different language of what it has said.]

§4. Some eminent thinkers have said that what leads us to use general names is the infinite multitude of individual objects; we can’t have a name for each, so we’re compelled us to make one name serve for many.

This is a very limited view of the function of general names. Even if we had a name for every individual object, we would still need general names as much as we now do. Without them we couldn’t express the result of a single comparison, or record any one of the uniformities existing in nature; and our inductions would hardly be in better shape than if we had no names at all. With only names of individuals—i.e. proper names—we. . . .couldn’t assert any proposition except the unmeaning ones formed by predicating two proper names one of another [e.g. ‘Cicero is Tully’]. It is only through general names that we can convey any information, predicate any attribute of an individual, let alone a class. We could in theory manage with no general names except the abstract names of attributes; all our propositions would then be of the form ‘Individual object X possesses attribute A’ or ‘Attribute A1 is always (or never) conjoined with attribute A2’. In fact, though, mankind have always given general names to objects as well as to attributes, and indeed before attributes; but the general names given to objects imply attributes, derive their whole meaning from attributes, and are chiefly useful as means for predicating the attributes that they (the general names) connote.

Now we must consider what principles are to be followed in giving general names so that the purposes of induction are best served by these names and the general propositions in which they occur.

NOTES

[1] [Mill has two footnotes to this paragraph; one quoting another writer saying the same thing; the other correcting someone who had thought that Mill, by what he wrote here about ‘association’, was committing himself to the view that ‘belief is nothing but an irresistible association’. Mill says:] I express no theory about the ultimate analysis of reasoning or of belief.

Chapter 4. What is needed for a philosophical language. The principles of definition

§1. [In this chapter and the next, ‘philosophical language’ means ‘language suitable for use in science’.] If we are to have a language perfectly suitable for investigating and expressing general truths, there are two main requirements (and several minor ones). (i) Every general name should have a meaning, steadily fixed, and precisely determined. When the names that we have are in this way fitted for the work they are to do, the next requirement (and the second in order of importance) is that (ii) we should have a name wherever one is needed—wherever there’s something to be designated by it that it is important to express. The present chapter will be entirely concerned with (i); ·I shall come to (ii) in chapter 6·.

§2. Every general name must have a certain and knowable meaning. Now the meaning of a general connotative name lies in the connotation [see Glossary]. . . . Thus, the name ‘animal’ being given to everything that has the attributes of •sensation and •voluntary motion, the word connotes those attributes exclusively, and they are the whole of its meaning. If the name is abstract, its denotation is the same as the connotation of the corresponding concrete name—it directly designates the attribute that the concrete term implies. [To make sure that that’s clear: The attribute of femininity has a name, an abstract word that denotes or directly designates it, namely ‘femininity’. The concrete word ‘woman’ implies or connotes that attribute, meaning that an item gets the name ‘woman’ because it has femininity. In that explanation, ‘woman’ could have been replaced by ‘feminine’; Mill’s category of ‘concrete’ terms includes adjectives as well as nouns.] To give a precise meaning to general names is to fix steadily the attribute(s) connoted by each concrete general name, and denoted by the corresponding abstract. Abstract names are created before concrete ones, as is proved by the fact that they are almost always derived from them; so we can consider the meaning of each as determined by and dependent on the meaning of the corresponding concrete name; so that the problem of •making our general language distinct [here = ‘clear’] is included in the problem of •giving a precise connotation to all our concrete general names.

This isn’t hard in the case of new names—technical terms created by scientific inquirers for the purposes of science or art [see Glossary]. It’s harder when when a name is in common use; the problem then is not that of electing a convenient connotation for the name but that of discovering and fixing the connotation it is already used with. That this can ever be a matter of doubt is a sort of paradox. But when the vulgar apply the same name to a number of different things they seldom know exactly what assertion they intend to make, what common property they mean to express. [Mill explains that he takes ‘the vulgar to include ‘all who don’t have accurate habits of thought’.]

When they apply a name to an object, all it expresses is a confused feeling of resemblance between that object and other things they have been accustomed to denote by that name. They have applied ‘stone’ to various objects; then they see a new object that appears to them somewhat like the previous ones, and they call it a ‘stone’, without asking themselves

•in what respect is this like the stones I have encountered previously? or

•for a thing to count as a ‘stone’, how closely must it resemble other stones?

They don’t know how the second question would be answered by the best authorities; they don’t even know how they themselves would answer it. But this rough general impression of resemblance is made up of particular features of resemblance; and it is the logician’s business to analyse it into them, i.e. to discover what points of resemblance among the different things commonly called ‘stones’ have given ordinary people this vague feeling of likeness that has dictated the uses of the word ‘stone’.

But though general names are applied by the vulgar on the basis of a mere vague resemblance, in due course people assert general propositions in which predicates are applied to all the things that are denoted by the name. Each of these propositions predicates some more or less vaguely conceived attribute; the ideas of these various attributes come to be associated with the name, and in a sort of uncertain way [Mill’s phrase] it comes to connote them; people hesitate to apply the name to anything that doesn’t have all the attributes commonly predicated of the class. In this way, the propositions that common minds are in the habit of hearing or uttering concerning a class make up in a loose way a sort of connotation for the class-name. Take the word ‘civilised’. Even among the most educated persons you won’t find many who would undertake to say exactly what ‘civilised’ connotes. Yet those who use the word feel that they’re using it with a meaning; and this meaning is made up, in a confused way, of everything they have heard or read about civilised men and civilised communities.

It’s probably at this stage in the career of a concrete name that the corresponding abstract name generally comes into use. Under the notion that the concrete name must convey a meaning, i.e. that there’s some property common to all the things it denotes, people give a name to this common property. From the concrete ‘civilised’ they form the abstract ‘civilisation’. But since most people have never considered the different things that are called by the concrete name, comparing them so as to ascertain what properties (if any) they have in common, each person is thrown back on the marks by which he himself has customarily been guided in his application of the term; and these, being merely vague hearsays and current phrases [Mill’s five-word phrase], are not the same in any two persons or in one person at different times. Hence the word that professes to designate an unknown common property conveys different ideas to almost any two minds. Think about ‘civilised’ again! No two persons agree about what items are civilised; and when something is called ‘civilised’, no-one else knows what he means to assert, and the speaker doesn’t know exactly. This uncertainty shows up even more strikingly with many other words—consider ‘honour’ and ‘gentleman’.

I hardly need to say that if no-one can tell exactly what the proposition P means, it can’t have been brought to the test of a correct induction. [It’s not clear what Mill means by that, is it? But at least it’s clear that he includes the statement that P can’t be used in a correct induction. He continues:] Whether a name is to be used as an instrument of thinking or as a means of communication, it is imperative to •determine exactly the attribute(s) that it is to express—i.e. to •give it a fixed and ascertained connotation.

§3. It would be a complete misunderstanding of a logician’s role in dealing with terms already in use if we were to think that

because a name doesn’t now have an ascertained connotation, anyone is free to give it such a connotation at his own choice.

The meaning of a term actually in use is not •an arbitrary quantity to be fixed but •an unknown quantity to be sought.

[Mill’s opening paragraph on this topic points out—at perhaps more length than is needed—the disadvantages of giving to a word in common use a meaning that creates a ‘rupture’ with its its present meaning. He concludes:] The fixed and precise connotation that the word receives should be. . . .in agreement (as far as it goes) with the vague and fluctuating connotation that the term already had.

To fix the connotation of a concrete name, or the denotation of the corresponding abstract name, is to define the name. When this can be done without conflicting with any generally accepted assertions, the name can be defined in accordance with its existing common use. Instead of ‘defining the name’, this is vulgarly called ‘defining the thing’, meaning ‘defining the ·relevant· class of things’—for nobody talks of defining an individual. What is meant by this improper way of talking is: define the name subject to the condition that it shall denote those things. This presupposes a comparison of the things, feature by feature and property by property, to discover what attributes they agree in; and often enough it also involves a strictly inductive operation to discover some unobvious agreement that is the cause of the obvious ones.

In order to give a connotation to a name, consistently with its denoting certain objects, we have to select from among the attributes in which those objects agree. So the first logical operation is to discover what they do agree in; and then the question arises: Which of these common attributes should we select to associate with the name? In many cases the common properties. . . .are extremely numerous. Our choice is narrowed down first by the preference to be given to properties that are well known and commonly predicated of the class; but even these are often too numerous to be all included in the definition, and anyway the most generally known properties may not be the best ones to mark out the class from all others. So we should select from among the common properties the ones (if there are any) on which it has been discovered by experience or proved by deduction that many others depend; or at least that are sure marks of many others. We thus see that to frame a good definition of a name already in use is a matter not of •choice but of •discussion; and discussion not merely about linguistic usage but also about the properties of things, and even about the origin of those properties. Every enlargement of our knowledge of the objects the name is applied to is liable to suggest an improvement in the definition. It is impossible to frame a perfect set of definitions on any subject until the theory of the subject is complete; and as science progresses so do its definitions.

§4. When the discussion of Definitions turns not on the use of words but on the properties of things, Whewell calls it ‘the explication of conceptions’; and the act of learning more about what detailed resemblances a classification is based on he calls—in his technical phraseology—unfolding the general conception in virtue of which those things are so classed. His terminology appears to me to have a darkening and misleading tendency, but several of his remarks are so good that I shall take the liberty of transcribing them.

He observes that many of the controversies that have loomed large in the formation of the existing body of science have had ‘the form of a battle of definitions’. He continues:

‘For example, the inquiry into the laws of falling bodies led to the question whether the proper definition of uniform force is that it generates a velocity proportional to the space or to the time from rest. The controversy about the vis viva was about the proper definition of the measure of force. A principal question in the classification of minerals is: what is the definition of a mineral species? Physiologists have tried to throw light on their subject by defining organisation, or some similar term.’

Questions of the same nature were long open, and are not yet completely closed, concerning the definitions of specific heat, latent heat, chemical combination, and solution. He goes on:

‘These controversies have never been questions of insulated and arbitrary definitions, as men seem often tempted to think they were. In all cases there’s a tacit assumption of some proposition that •is to be expressed by means of the definition and •gives it its importance. The dispute about the definition thus acquires a real value, and becomes a question of true and false. In the discussion of the question “What is a uniform force?” it was taken for granted that gravity is a uniform force. In the debate on the vis viva it was assumed that in the interaction of bodies the whole effect of the force is unchanged. In the zoological definition of species (that it consists of individuals that have or may have come from the same parents), it is presumed that individuals so related resemble each other more than those that are excluded by such a definition; or perhaps that species so defined have permanent and definite differences. A definition of organisation or any other term that wasn’t used to express some principle would have no value.

‘So the establishment of a right definition of a term can be a useful step in the explication of our conceptions, but only when we are thinking of some proposition in which the term is used. For then the question really is: how must we understand and define the conception so that the proposition comes out true?

‘To unfold our conceptions by means of definitions has never been serviceable to science, except when it has been associated with an immediate use of the definitions. The attempt to define uniform force was combined with the assertion that gravity is a uniform force; the attempt to define accelerating force was immediately followed by the doctrine that accelerating forces can be compounded; the process of defining momentum was connected with the principle that momenta gained and lost are equal; naturalists would have given in vain the definition of species that I have quoted if they hadn’t also given the characters of species so separated. . . Definition may be the best way of explaining our conception, but there would be no point in explaining it in any way if it weren’t to be used in expressing truth. When a definition is propounded to us as a useful step in knowledge, we are always entitled to ask what principle it serves to enunciate.’

In giving, then, an exact connotation to the phrase ‘uniform force’ it was understood that the phrase should continue to denote gravity. So the discussion regarding the definition came down to the question: What uniformity is there in the motions produced by gravity? By observations and comparisons it was found that what was uniform in those motions was •the ratio of the velocity acquired to •the time elapsed, equal velocities being added in equal times. A uniform force was therefore defined as a force that adds equal velocities in equal times. Similarly in defining momentum: it was already an accepted doctrine that when two objects collide the momentum lost by one is equal to that gained by the other. It was thought necessary to preserve this proposition because it was felt to contain a truth; even a superficial view of the phenomena left no doubt that in any collision there was something of which one body gained precisely the amount that the other lost; and the word ‘momentum’ had been invented to express this unknown something. (Sometimes a definition is required to preserve a proposition that is firmly fixed in popular belief; but not in this case, because the proposition in question had never been heard of by any but the scientifically instructed.) What was needed to settle the definition of momentum, therefore, was the answer to the question: When one body sets another body in motion, it loses exactly as much what as the other gains? When experiments had shown that the answer is ‘the product of the velocity of the body by its mass or quantity of matter’, this became the definition of momentum.

The following remarks of Whewell’s are therefore perfectly correct:

‘The business of definition is part of the business of discovery. . . To make a definition that has any scientific value one needs a great deal of acuteness in discovering the truth. . . When it’s clear to us what ought to be our definition, we know pretty well what truth we have to state. . . . The writers on logic in the middle ages made definition the last stage in the progress of knowledge; and this view of theirs, at least, is confirmed by the history of science and the philosophy derived from that history.’

For in order to judge finally how the name that denotes a class may best be defined, we must know all the properties common to the class, and all the relations of causation or dependence among those properties.

The most felicitous kind of definition is the one where the properties that are fittest to be selected as marks of other common properties are obvious and familiar, especially if they also contribute greatly to giving people the general impression of resemblance that led to the formation of the class in the first place. But a class often has to be defined by some property that isn’t familiarly known, though it needs to be the best mark of those that are ·familiarly· known. De Blainville, for instance, based his definition of life on the process of decomposition and recomposition that continually occurs in every living body, so that the particles composing it are never for two instants the same. This is not one of the most obvious properties of living bodies; an unscientific observer might miss it altogether. Yet great authorities. . . . have thought that no other property satisfies so well the conditions required for the definition.

§5. Having set out the principles that we ought usually to follow when we are trying to give a precise connotation to a term in use, I must now add that it isn’t always practicable to stick to those principles, and that even when it is practicable it isn’t always desirable.

There are many cases where it’s impossible to comply with all the conditions of a precise definition of a name in agreement with usage. In many cases a word W can’t be given one connotation that will make W denote everything that is customarily denotes, or that makes true all the propositions that it customarily enters into and that have any foundation in truth. Independently of accidental ambiguities, where the different meanings have no connection with one another, it often happens that a word is used in two or more senses derived from each other yet radically distinct. So long as a term is vague—i.e. doesn’t have a permanently fixed connotation—it is constantly liable to be applied by extension from one thing to another, until it reaches things which have little if any resemblance to those that were first designated by it.

[Mill quotes Dugald Stewart, a Scottish philosopher and mathematician, describing this process as happening along a series of objects so that they all fall under a single general name although no three of them have anything in common and the first is so unlike the last that] ‘no stretch of imagination can conceive how thoughts were led from one to the other. Yet the transitions may have been all so easy and gradual that if they were successfully detected by a theorist, we would instantly recognise not only the likelihood but the truth of the conjecture ·about how a single general word travelled along the series·; in the same way that we are confidently certain of the well-known etymological process that connects the Latin preposition e or ex with the English noun “stranger” as soon as the intermediate links of the chain—e, ex, extra, “extraneous”, étranger, “stranger”—are present for our examination.’

Stewart uses the adjective ‘transitive’ for the applications that a word acquires through this gradual extension of it from one set of objects to another. After briefly illustrating ones that are the result of local or casual associations, he proceeds as follows:

‘But although most of the transitive or derivative applications of words depend on casual and unaccountable whims of the feelings or the imagination, there are certain cases where they open a very interesting field of philosophical speculation. They are the ones in which an analogous transference of the corresponding term can be seen in many or most other languages, this being a uniformity that must be ascribed to the essential forces of human nature. But even in these cases the explanation doesn’t usually lie in similarities amongst the objects the word is applied to; it often comes from associations of ideas based on the common faculties, common organs, and common condition of the human race. . . The resulting meaningpattern will vary according to how intimate and how strong the underlying associations are. Where the association is slight and casual, the various meanings will remain distinct from each other, and will often come to look like capricious varieties in the use of the same arbitrary sign. Where the association is so natural and habitual as to become virtually unbreakable, the transitive meanings will coalesce in one complex conception; and every new transition will become a more comprehensive generalisation of the term in question.’

I call attention to the law of mind expressed in that last sentence; it’s the source of the perplexity so often experienced in detecting these transitions of meaning. Ignorance of that law is the shoal on which some of the most powerful intellects that have adorned the human race have been stranded. Plato’s inquiries into the definitions of some of the most general terms of moral theory are described by Bacon as being closer to a true inductive method than any of the other ancients achieved; and they are indeed almost perfect examples of the preparatory process of comparison and abstraction; but because he didn’t know of the law just mentioned, he often wasted the powers of this great logical instrument on inquiries in which it couldn’t produce any result because the phenomena whose common properties he so elaborately tried to detect didn’t have any common properties. Bacon himself fell into the same error in theorising about the nature of heat, in which he jumbled together—under the name ‘hot’—classes of phenomena that have no property in common. . . . Aristotle and his followers were well aware of ambiguities in language, and delighted in distinguishing them. But they never suspected ambiguity in the cases where (as Stewart remarks) the association from which the transition of meaning arose is so natural and habitual that the two meanings blend together in the mind, and a •real transition becomes an •apparent generalisation. They took endless trouble trying to find a definition that would serve for several distinct meanings at once; as in an instance that Stewart mentions, the meaning of ‘causation’. The corresponding Greek word prompted them to look for the the common idea that runs through an effect’s efficient cause, material cause, formal cause, and final cause [see Glossary on efficient]. Stewart adds: ‘Other philosophers have produced idle generalities about the ideas of the good, the fit, and the becoming—all this arising from the same undue influence of popular epithets on the speculations of the learned.’

Stewart considers ‘beautiful’ to be one of the words that have undergone so many transitions of meaning that there’s no longer any trace of a property possessed by all and only the objects the word is applied to. And I can’t help feeling (though this isn’t a question in logic!) a considerable doubt whether the word ‘beautiful’ connotes the same property when we speak of

•a beautiful colour,

•a beautiful face,

•a beautiful scene,

•a beautiful character, and

•a beautiful poem.

No doubt the word was extended from one of these objects to another on account of a resemblance between them, or (more likely) between the emotions they aroused; and, by this progressive extension it has at last reached things very remote from the visible objects that it was originally applied to. It is at least questionable whether the things that would ordinarily be called ‘beautiful’ have any property in common except agreeableness. The term certainly does connote this, but that can’t be all that people usually mean by ‘beautiful’ because many agreeable things are never called beautiful. If that is right, the word ‘beautiful’ can’t be given any fixed connotation that will make it denote all and only the things that it now denotes in common use. But it ought to have a fixed connotation; as long as it doesn’t it is unfit to be used as a scientific term and is a perpetual source of false analogies and erroneous generalisations.

This illustrates my remark that even when there is a property common to all the things denoted by a name, it’s not always desirable to make that property the definition and exclusive connotation of the name. The various things called ‘beautiful’ are certainly alike in being agreeable; but to make this the definition of beauty, and thus extend ‘beautiful’ to all agreeable things, would be •to drop a portion of meaning that the word really though unclearly conveys, and •to contribute to making ourselves overlook and forget those qualities of the objects that the word previously, though vaguely, pointed at. In such a case where we want to give a fixed connotation to a term, it is better to do this by restricting its use than by extending it. It is better •to exclude from the ·range of the· adjective ‘beautiful’ some things that it is commonly applied to than •to leave out of its connotation any of the qualities that may—even if they are occasionally lost sight of—have guided people’s minds in the commonest and most interesting applications of the term. When people call anything ‘beautiful’ they certainly think they are saying more than merely that it is agreeable. They think they’re ascribing a special sort of agreeableness, analogous to what they find in some other of the things that they are also accustomed to calling ‘beautiful’. So if there is any special sort of agreeableness that is common to (if not all, at least) the principal things that are called ‘beautiful’, limiting the denotation of the term to those things is better than leaving that kind of quality without a word to connote it, thereby diverting attention from its special features.

§6. Here is a rule of terminology that is of great importance though it has hardly been recognised as a rule except by a few thinkers of the present century:

In trying to rectify the use of a vague term by giving it a fixed connotation, take care not to discard any portion of the connotation that the word previously had, however unclearly; unless you are doing this advisedly, on the basis of a deeper knowledge of the subject.

Otherwise language loses one of its inherent and most valuable roles, as conservator of ancient experience, keeper-alive of thoughts and observations of former ages that may be alien to the tendencies of the passing time. This function of language is so often overlooked or undervalued that a few observations on it appear to be extremely required.

There’s a constant tendency for any word, through familiar use, to lose some of its connotation; this happens even when its connotation has been fixed, but still more if it has been left as a vague unanalysed feeling of resemblance. It is a well-known law of the mind that

a word originally associated with a very complex cluster of ideas doesn’t call up all those ideas in the mind every time it is used; it calls up only one or two, from which the mind runs on by fresh associations to another set of ideas without waiting for the suggestion of the ·omitted· remainder of the complex cluster.

If this weren’t so, our processes of thought couldn’t be anywhere near as fast as they are. Very often, indeed, when we’re using a word in our mental operations we run on to new trains of ideas by other associations that the mere word starts off, without having brought into our our imagination any part whatever of the complex idea corresponding to the meaning of the word. When that happens we are using the word in an almost mechanical manner, which we can do even when we using it well and accurately in carrying on important processes of reasoning. (Some metaphysicians, generalising from the extreme instances of this, have fancied that all reasoning is merely the mechanical use of a set of terms according to a certain form.) We can discuss and settle the most important interests of towns or nations by applying general theorems or practical maxims previously laid down, without giving any thought to the houses and green fields, the busy market-places and domestic hearths, that those towns and nations consist of and are part of what ‘town’ and ‘nation’ mean.

Since general names come in this way to be used (and even to do some of their work well) without suggesting to the mind their whole meaning, and often suggesting only a very small part or no part of that meaning, it’s not surprising that these words become unable to suggest any of ideas belonging to them except those with which the association is •most immediate and strong, or •most kept up by the incidents of life. . . . Words naturally retain much more of their meaning to persons of active imagination, who habitually represent things to themselves with the detail that belongs to them in the actual world. For minds of a different kind, the only antidote to this corruption of language is to have a habit of using the name in connection with all the properties that it originally connoted, thus keeping up the association between the name and those properties.

But it can’t do this unless the predicates retain their association with the properties they connote. The propositions can’t keep the meanings of the words alive if their meanings die! It often happens that propositions are mechanically repeated, mechanically held in the memory, and their truth confidently assented to and relied on, while •they bring no meaning clearly to the mind, and •the matter of fact or law of nature that they originally expressed is lost sight of and plays no part in the person’s thinking. In subjects that are both familiar and complicated—especially moral and social subjects—it is commonly noticed that many important propositions are believed and repeated from habit by people who couldn’t say what they mean and whose speech and other conduct aren’t affected by their supposed truth. That is why the traditional maxims of old experience, though seldom questioned, often have so little effect on the conduct of life; it’s because most people don’t really feel their meaning until personal experience brings it home to them. It is also why so many doctrines of religion, ethics, and even politics, so full of meaning and reality to first converts. . . ., show a tendency to degenerate rapidly into lifeless dogmas, a tendency that all the efforts of an education expressly and skillfully directed to keeping the meaning alive are barely sufficient to counteract.

. . . .It is natural and inevitable that in every age a certain portion of our recorded and traditional knowledge, not being continually suggested by the pursuits and inquiries mankind are at that time absorbed in, should fall asleep, as it were, and fade from the memory. What saves it from being totally lost is this:

The propositions or formulas arising from previous experience still remain; they are only forms of words, but of words that once had a meaning and are still supposed to do so; and this (suspended) meaning can be historically traced, and may be recognised by sufficiently able minds as still being matter of fact, or truth.

While the formulas remain, the meaning may at any time revive. . . .and be announced to mankind not as a discovery but as the meaning of something that they have been taught and still profess to believe.

Thus there’s a perpetual oscillation in spiritual truths, and in spiritual doctrines of any significance even if they aren’t true: their meaning is almost always in a process either of being lost or of being recovered. Look at the history of mankind’s more serious convictions—the opinions by which they do (or think they should) regulate their lives—and you’ll see that even when recognising verbally the same doctrines, they differ through the years in how much meaning, and even what kind of meaning, to attach to them. [Mill repeats that each age drops the parts of the meaning that don’t concern it; but ‘any mind duly prepared’ can recover the lost meaning, revive it, and get it back into the common meaning.]

The arrival of this satisfactory upshot can be significantly delayed by the shallow conceptions and incautious proceedings of mere logicians. It happens like this:

When a word W has lost part of its significance and hasn’t yet begun to get it back, persons arise whose favourite idea is the importance of clear conceptions and precise thought and thus the necessity of definite language. When they examine the old formulas containing W they easily sere that it is used in them without any meaning; and if they aren’t capable of rediscovering the lost meaning, they naturally enough dismiss the formulas and define W without reference to them.

In doing this they •fasten W down to what it connotes in common use at the time when it conveys the smallest quantity of meaning, and •introduce the practice of consistently and uniformly using it according to that connotation. In this way W acquires a much wider denotation than it had before; it becomes applicable to many things to which it was previously refused, and that earlier refusal now looks arbitrary. Of the propositions in which W was formerly used, those that were true because of the forgotten part of its meaning are now, by the clearer light shining from the definition, seen not to be true according to the definition; yet the definition is the recognised and sufficiently correct expression of all that is in the mind of anyone who uses W at the present day. The ancient formulas are thus treated as prejudices; and people are no longer taught to believe that there is truth in them. In the general mind they are no longer surrounded by respect, and ready at any time to suggest their original meaning. Whatever truths they contain are rediscovered far more slowly, and when they are rediscovered they look like novelties, and that in some degree at least counts against them.

Here is an example. The minds of thinking persons have always been concerned with the question ‘What is virtue?’ or ‘What is a virtuous character?’ (except where moral thinking has been suppressed by outward compulsion, and where the feelings that prompt it are still satisfied by the traditional doctrines of an established faith). Of the different answers that have at various times met with some acceptance each has been a perfect mirror-image of the age that gave it birth. One answer was this: virtue consists in correctly calculating our own personal interests, either in this world only or also in another world. To make this theory plausible, it was of course necessary that the only beneficial actions that people in general were accustomed to see. . . .were results of a prudential concern for self-interest, or at least could plausibly be supposed to be so; with the result that in ordinary usage the words connoted no more than was set down in the definition.

Suppose now that the friends of this theory managed to introduce a consistent and regular use of ‘virtue’ according to this definition. Suppose that they succeeded in •banishing the word ‘disinterested’ [see Glossary] from the language, and •driving out of the language all expressions frowning on selfishness or commending self-sacrifice or implying that generosity or kindness is anything but giving a benefit in order to receive a greater personal advantage in return. The flouting of the old formulas for the sake of preserving clear ideas and consistency of thought would obviously be a great evil. . . .

[The next sentence refers to Samuel Taylor Coleridge—poet, critic, and philosopher.] The Coleridge school hold that

the language of any people who have a long history of culture is a sacred deposit, the property of all ages—something that no one age should consider itself empowered to alter.

Put like that it is exaggerated; but it is based on a truth that is frequently overlooked by the class of logicians who •think more of having a clear meaning than of having a comprehensive one, and who •see that every age is adding to the truths that it has received from its predecessors, but •fail to see that a contrary process of losing truths already possessed is also constantly going on isn’t easy to counteract. Language is the depository of the accumulated body of experience to which all former ages have contributed and which is the inheritance of all yet to come. . . . However much we may be able to improve on the conclusions of our forefathers, we ought to be careful not to let any of their premises slip through our fingers. It may be good to alter the meaning of a word, but it is bad to let any part of the meaning drop. Anyone wanting to introduce a more correct use of a word W that has important associations should be required to possess an accurate acquaintance with the history of W and of the opinions that in different ages it served to express. To be qualified to define W we must know all that has ever been known of the properties of the class of objects which are, or originally were, denoted by it. If we give it a meaning according to which some proposition that was once generally held to be true comes out false, we ought to be sure that we know how W was understood by those who believed the proposition.

Chapter 5. The natural history of the variations in the meaning of terms

§1. I have described just one way in which words in common use are liable to shift their connotation. The truth is that the connotation of such words is perpetually varying ·in other ways too·, as might be expected given how they acquire their connotation in the first place. A technical term invented for purposes of art or science possesses from the outset the connotation given to it by its inventor; but a word W that is in everyone’s mouth before anyone thinks of defining it gets its connotation only from the facts that are habitually brought to mind when W is used. Looming large among these facts are the properties common to the things denoted by W; they would be the whole story if language were regulated by convention rather than by custom and accident. But besides these common properties, which if they exist are certainly present whenever W is used, any other circumstance [see Glossary] may be found along with it, •casually but •often enough to become associated with W in the same way, and as strongly, as the common properties themselves. As this association develops, people give up using W in cases where those casual circumstances don’t exist. They prefer using some other word, or W with some qualifier rather than using an expression that will call up an idea they don’t want to arouse. The originally casual circumstance thus becomes regularly a part of the connotation of the word.

This continual incorporation of accidental circumstances into the permanent meanings of words has two upshots worth noting. (i) There are very few exact synonyms. (ii) As everyone knows, the dictionary meaning of a word is a very imperfect account of its real meaning. The dictionary meaning is marked out in a broad, blunt way, and probably includes everything that was originally required for the correct use of the word; but in the course of time so many extra associations stick to words that anyone who tried to use them with no guide except the dictionary would muddle up a thousand little distinctions and subtle shades of meaning that dictionaries ignore. We see this in the conversation or writing of a foreigner who isn’t thoroughly master of the language. The history of a word, by showing the causes that determine its use, is in these cases a better guide to its use than any definition; for definitions can only show the word’s meaning at the particular time, or at most its series of meanings through time, but its history can show the law the series was governed by. For example, a dictionary would be no guide to the correct use of the word ‘gentleman’. Originally it meant simply a man born in a certain rank. From this it gradually came to connote all the qualities or casual circumstances that were usually found to belong to persons of that rank. This at once explains why in one of its common meanings it means •anyone who doesn’t have to work for a living, in another •anyone who doesn’t have to do manual labour for a living, and in its more elevated meaning it has at every time signified •anyone whose conduct, character, habits, and outward appearance were—at that time—typical of (or thought to be typical) of persons born and educated in a high social position.

It often happens that of two words whose dictionary meanings are the same or only slightly different, W_1 is the proper word to use in one set of circumstances and W_2 in another set, though we can’t show how the custom of so using them originally arose. The accident that W_1 and not W_2 was used on a particular occasion or in a particular social circle will create such a strong association between W_1 and some specialty of circumstances [Mill’s phrase] that mankind abandons the use of it in any other case, and the specialty becomes part of its meaning. The tide of custom first drifts W_1 onto the shore of a particular meaning, then retires and leaves it there.

[Mill gives the example of the word ‘loyalty’, which first meant ‘fair, open dealing, and fidelity to engagements’ and now means ‘fidelity to the throne’. He doesn’t know how the change came about, but offers a guess.]

§2. In many cases a circumstance that at first casually came into the connotation of a word that originally had no reference to it eventually comes to supersede the original meaning and becomes not merely a part of the connotation but the whole of it. An example is the word ‘pagan’, paganus [Latin]. This was originally equivalent to ‘villager’—the inhabitant of a pagus or village. At a particular era in the spread of Christianity over the Roman empire the •adherents of the old religion were nearly the same group of individuals as •the villagers or country people, because the inhabitants of the towns were the earliest converted to Christianity. . . . From this casual coincidence the word paganus carried with it, and began ever more steadily to suggest, the idea of someone who worships the ancient divinities; until at last it suggested that idea so forcibly that people who didn’t want to suggest that idea avoided using that word. But when paganus had come to connote heathenism, the very unimportant detail about not living in a city was soon disregarded in the use of the word. Because there was seldom any need to say something about heathens who lived in the country there was no need for a separate word to denote them; so ‘pagan’ came not merely to include ‘heathen’ in its meaning but to mean that exclusively.

Another example is the word ‘villain’ or ‘villein’. In the middle ages this term had a connotation as strictly defined as a word could have, being the proper legal label for persons who were the subjects of the less burdensome forms of feudal bondage. The scorn of the semi-barbarous military aristocracy for these abject dependants of theirs made the act of likening someone to this class of people a mark of the greatest contempt; and that same scorn led them to ascribe to the same people all sorts of hateful qualities. . . . These circumstances combined to attach to the term ‘villain’ ideas of crime and guilt so forcibly that the application of the epithet even to those to whom it legally belonged became an insult and was abstained from whenever no insult was intended. From that time guilt was part of the connotation; and it soon became the whole of it because mankind had no urgent reason to continue using their language to distinguish •bad men low down on the social scale from •bad men of any other rank in life.

Examples like these, where the original meaning of a term is totally lost—another and an entirely distinct meaning first being grafted onto the former, and eventually replacing it—present examples of the double movement that is always occurring in language: two counter-movements,

•one of generalisation, by which words are constantly losing portions of their connotation, and coming to have less meaning and ·therefore· more general acceptation;

•the other of specialisation, by which words—perhaps even the same ones—are continually taking on fresh connotation, acquiring additional meaning by being restricted in their range of application.

This double movement is important enough in the natural history of language. . . .to justify dwelling a little longer on its nature and causes.

§3. To begin with the movement of generalisation. It might seem unnecessary to dwell on meaning-changes that arise merely from a word’s being used in a looser and wider sense than belongs to it, by ignorant people who haven’t properly mastered its accepted connotation. But this ·is an important topic, because it· is a real source of alterations in the language: if a word W is often used in cases where one of the qualities it connotes doesn’t exist, it stops suggesting that quality with certainty; then even those who know the proper meaning of W prefer to express that meaning in some other way, leaving the original word to its fate. . . . This gives us insight into the way languages have degenerated at times when literary culture was suspended; and we’re now in danger of experiencing a similar evil through the superficial extension of that same culture. [The next sentence is verbatim from Mill.]

So many persons without anything deserving the name of ‘education’ have become writers by profession that written language may almost be said to be principally wielded by persons ignorant of the proper use of the instrument, and who are spoiling it more and more for those who understand it.

Vulgarisms, which creep in nobody knows how, are daily depriving the English language of valuable ways of expressing thought. The verb ‘transpire’ used to mean ‘become known through unnoticed channels’—to exhale, as it were, into the public arena like a vapour or gas. But recently people have started to use this word as a supposedly more decorative synonym of ‘happen’: ‘the events which have transpired in the Crimea’, meaning the incidents of the war. This vile specimen of bad English is already seen in the dispatches of noblemen and viceroys; and before long, it seems, nobody will understand ‘transpire’ if used in its proper sense. Some words come to be used in senses unknown to genuine English not because the writers love decoration but simply because they are uneducated. The use of ‘aggravating’ for ‘annoying’, in my boyhood a vulgarism of the nursery, has crept into almost all newspapers, and into many books; and when the word is used in its proper sense [namely, ‘making worse’], as when writers on criminal law speak of ‘aggravating’ and ‘extenuating’ circumstances, they are probably misunderstood, even today. These corruptions of language do harm. Those who are struggling to express themselves clearly with precision (knowing from experience how hard it is to do this) find their resources continually narrowed by illiterate writers who seize and twist from its purpose some word or phrase that once served to convey briefly and compactly an unambiguous meaning. [Mill then offers a page full of further examples and further protests.]

There is also a tendency for the meanings of names to be generalised not because of ignorant misuse but because the number of names we have doesn’t keep up with the number of things we want to talk about. It is usually very difficult to bring a new name into use (except for new scientific terminology that laymen don’t meddle with). And quite apart from that it’s natural to prefer giving a new object a name that at least expresses its resemblance to something already known, because an entirely new name would at first convey no information. [Mill gives examples: ‘salt’, ‘oil’, ’glass’, ’soap’. He then says that this kind of language-change happens even more with words that express the complicated phenomena of mind and society. Historians, travellers, and others who also speak or write about moral and social phenomena that they aren’t closely acquainted with are the great agents in this process. Such people (apart from those who are unusually well-educated and thoughtful) have an eminently scanty vocabulary for such subjects. They have a small set of words that they are accustomed to and that they use to describe widely different phenomena, because they have never sufficiently analysed the facts those words correspond to in their own country to have attached perfectly definite ideas to the words. The first English conquerors of Bengal, for example, took the phrase ‘landed proprietor’ into a country where the rights of individuals over the soil were extremely different in degree and even in kind from those recognised in England. Applying the phrase, with all its English associations, in Bengal:l

•to one who had only a limited right they gave an absolute right,

•from one who didn’t have an absolute right they took away all right,

•whole classes of people were driven to ruin and despair, and

•the whole country was filled with banditti, creating a feeling that nothing was secure.

In this way, with the best of intentions, they produced more disorganisation of society than had been produced by the most ruthless of its barbarian invaders. . . .

§4. Two opposite processes go on together: (i) Names come to be applied to more things because the rapid growth of ideas ·of things· outstrips the growth of names. (ii) names become on the contrary restricted to fewer occasions, by acquiring extra connotation from circumstances that weren’t originally in the meaning but have become connected with it in the mind by some accidental cause. We saw in the words ‘pagan’ and ‘villain’ remarkable examples of •word-meanings that became specialised because of casual associations, as well as •meaning-generalisations in new directions that often follow this.

Similar specialisations have occurred often in the history even of scientific terminology. [After presenting many examples of this, all contained in a long quotation from J. A. Paris’s Pharmacologia, Mill continues:]

A generic term is always liable to become limited in this way to a single species, or even a single individual, if people have occasion to think and speak of that species or individual much oftener than of anything else contained in the genus. Thus

•for a stage-coachman ‘cattle’ are horses;

•for agriculturists ‘beasts’ are oxen; and

•for some sportsmen ‘birds’ are all partridges.

The law of language that operates in these trivial instances was also at work when Christianity named the single object of its worship by borrowing the words Theos [Greek], Deus, and ‘God’ from Polytheism. Almost all the terminology of the Christian Church is made up of words originally used in with much more general meanings. [He lists 14 examples, and adds:] It would be interesting to trace the process through which ‘author’ came in its most familiar sense to mean ‘writer’. . . .

Our ideas of pleasure and pain—and of things that we always think of as sources of our pleasures or pains—are the most liable to cling by association to anything they have ever been close to. So the extra connotation that a word most quickly and easily takes on is that of agreeableness or painfulness in their various kinds and degrees:

•good or bad thing;

•desirable or to be avoided;

•object of admiration/hope/love or hatred/dread/contempt.

Almost every word that •expresses a moral or social fact and •is apt to arouse strong favourable or hostile feelings carries with it, decidedly and irresistibly, a connotation of those strong feelings or at least of approval or disapproval. So that if you use any of those names in conjunction with others expressing the opposite feelings you’ll produce the effect of a paradox or even a contradiction in terms. A connotation acquired in this way has a poisonous effect on prevailing habits of thought, especially in morals and politics, as Bentham has often pointed out. It gives rise to the fallacy of ‘question-begging [see Glossary] names’. We’re inquiring whether a thing x possesses a property P or not, but P has become so associated with the name of x as to be part of its meaning, so that by merely uttering the name we assume the conclusion we were looking for; most apparently self-evident propositions are like that.

. . . .The logician, faced with such changes of meaning, should submit to them with a good grace once they have settled into place and can’t be dislodged; and if a definition is needed, define the word according to its new meaning, retaining the older as a second meaning in case of need, if there’s any chance of being able to preserve it either in the language of philosophy or in common use. Logicians can’t make the meanings of any but scientific terms; the meanings of all other words are made by the collective human race. But logicians can get a clear view of whatever it is that has surreptitiously guided the general mind to a particular use of a name; and when they have found it they can clothe it in such distinct and permanent terms that mankind will see the meaning which before they only felt, and from then on they won’t let it be forgotten or misapprehended.

Chapter 6. The principles of a philosophical language further considered

§1. Up to here I have discussed only one of the requirements of a language adapted for the investigation of truth, namely that each of its terms shall have a determinate and unmistakable meaning. There are other requirements, one of which is fundamental and nearly as important—if not as important—as the quality I have discussed at such length. For a language to be fitted for its purposes, it should be the case not only that •every word perfectly expresses its meaning but also that •every important meaning has its word. Whatever we have occasion to think of often, and for scientific purposes, ought to have a name assigned to it.

This requirement of philosophical language involves three conditions, which I’ll discuss in the next three sections.

§2. First, there ought to be all the names needed for recording individual observations in such a way that the words of the record show exactly what fact it is that has been observed. In other words, the language should have an accurate descriptive terminology.

The only things we can observe directly are our own sensations or other feelings, so a complete descriptive language would contain a name for every kind of elementary sensation or feeling. Combinations of these can always be described if we have a name for each of the elementary feelings that compose them; but it is a big help if the language has distinctive names not only for the elements but also for every combination of elements that recurs frequently. Why? Because this makes for brevity and thus for clearness, which often depends very much on brevity. On this topic I can’t do better than quote some of the excellent things Whewell has said about it:

‘The meaning of [descriptive] technical terms can be fixed in the first instance only by convention, and can be made intelligible only by presenting to the senses whatever it is that the terms are to signify. The knowledge of a colour by its name can only be taught through the eye.’ [Whewell then says that when a descriptive term T is explained in terms of a sense-presentation, that is what the learner must be reminded of by T—not any meaning that T or some part of it has in common language. If someone is introduced to ‘apple-green’ as the name of a kind of sensation, his later uses of the label should refer back to that sensation, avoiding irrelevant thoughts about what apples usually look like. Whewell continues:]

‘It is most important to remember this in connection with the •simpler properties of bodies such as colour and shape, but it’s equally true with for •more compound notions. In all cases the term is fixed to a particular meaning by convention; and a student wanting to use the word must be completely familiar with the convention, so that he has no need to rely on risky guesses based on the word itself.’ [He gives the example of ‘papilionaceous’, an adjective that marks off a kind of flower that resembles a butterfly [Latin papilio = ‘butterfly’]; but the conventional meaning brings in only a very small selection of butterfly-features; and a user of the word has to know those, i.e. to know the convention.]

In these cases where the thing named is a combination of simple sensations, the meaning of the name W can be learned without going back to the sensations themselves; the learner may get W’s meaning through other words—which is to say that W can be defined. The names of elementary sensations, or elementary feelings of any sort, can’t be defined; the only way to make their meaning known is to •make the learner experience the sensation or to •direct his thought somehow to his memory of having experienced it before. Hence the only things that can be handled in an exact descriptive language are the impressions on the outward senses and inward feelings that are connected with outward objects in a very obvious and uniform way. It would be useless to try to name the countless variety of sensations arising from diseases or from special physiological states: no-one can judge whether my sensation is the same as his, so there can’t be a name for it that has real community of meaning between the two of us. This also holds to a considerable extent for purely mental feelings. But in some of the sciences dealing with external objects the scientific language has been developed to a level of perfection that it’s scarcely possible to improve on.

[Mill now quotes a long passage in which Whewell writes almost rapturously about the basic descriptive means of botany, which build on fundamental names for parts of plants, the parts being identified through their role in the life of the plant. The quoted passage ends thus:] ‘Other characters. . . .are also conveyed with similar precision: colour by means of a classified scale of colours. . . This was done with most precision by Werner, whose scale of colours is still the most usual standard of naturalists. Werner also introduced more exact terminology for other characters that are important in mineralogy, e.g. “lustre”, “hardness”. But Mohs went further, with a numerical scale of hardness in which talc is 1, gypsum 2, calc spar 3, and so on. . . .’

§3. So much for language needed for recording our observation of individual instances. But when we proceed from observation to induction, or rather to the comparison of observed instances that is the preparatory step toward induction, we come to a second requirement, ·namely general names for properties that are likely to be central in high-level theories·.

Sometimes for purposes of induction we have to introduce some new general conception (Whewell’s phrase); that is, sometimes the comparison of a set of phenomena leads us to recognise in them some common feature that is to us a new phenomenon because this is the first time our attention has been directed to it. Whenever this happens it’s important to give a name to this new conception, this new result of abstraction; especially if the feature it involves •leads to many consequences or •is likely to be found also in other classes of phenomena. In most cases the meaning might be conveyed by combining several words already in use. But when a thing has to be often spoken of, there are more reasons than merely the saving of time and space for speaking of it in the most concise manner possible. What darkness would be spread over geometrical demonstrations if wherever the word ‘circle’ is used the definition of circle were used instead. In mathematics and its applications, where the nature of the processes demands that the •attention should be strongly concentrated. . . ., the importance of concentration also in the •expressions has always been felt; and a mathematician no sooner finds that he will often want to speak of the same two things together than he at once creates a term to express them whenever combined; . . . . not solely to shorten his formulae but also to simplify the purely intellectual part of his operations by enabling the mind to focus on the relation between the quantity S and the other quantities that enter into the equation, without being distracted by thinking unnecessarily of the parts of which S is itself composed.

But there’s also another reason for giving a brief and compact name to each of the more considerable results of abstraction that come up in our intellectual procedures. By naming them, we fix our attention on them; we keep them more constantly before the mind. The names are remembered, and being remembered they suggest their definition; whereas if instead the meaning had been expressed by putting together a number of other names, that particular combination of

words already in common use for other purposes

would have had nothing to make itself remembered by. If we want to make a particular combination of ideas permanent in the mind, nothing clinches it like a name specially devoted to expressing it. If mathematicians had had to speak of

•‘that to which a quantity, in increasing or diminishing, is always approaching nearer, so that the difference becomes less than any assignable quantity, but to which it never becomes exactly equal’

instead of expressing all this by the simple phrase

•‘the limit of a quantity’

we would probably have long remained without most of the important truths that have been discovered through the relation between quantities of various kinds and their limits. If physicists had had to speak of

•‘the product of the number of units of velocity in the velocity by the number of units of mass in the mass’

instead of speaking of

•‘momentum’

many of the truths of dynamics now grasped by means of this complex idea would probably have escaped notice because the idea itself wouldn’t have come to mind with sufficient readiness and familiarity. And on subjects closer to the topics of popular discussion, if you want to draw attention to some new or unfamiliar distinction among things you won’t find a better way to do it than by inventing or selecting suitable names for the special purpose of marking it.

A volume devoted to explaining what the writer means by ‘civilisation’ won’t convey as vivid a conception of it as will the single expression: Civilisation is a different thing from Cultivation. The compactness of that brief designation for the contrasted quality is an equivalent for a long discussion. If we want to impress forcibly on the understanding and memory the distinction between the two different conceptions of a representative government, we can’t do it better than by saying that Delegation is not Representation. Hardly any original thoughts on mental or social subjects ever make their way among mankind, or get their proper importance in the minds even of their inventors, until aptly-selected words or phrases have nailed them down (as it were) and held them fast.

§4. Now we come to the third requirement for a philosophical language, ·namely a name for every Kind·. By a Kind, you will remember, I mean

a class that is distinguished from all others not by one or a few definite properties, but by an unknown multitude of them; the combination of properties on which the class·ification· is based being a mere pointer to indefinitely many other distinctive attributes.

The class horse is a Kind, because the things that agree in having the characters by which we recognise a horse agree in a great many other properties that we know about and also—it can’t be doubted—in many more than we know. Animal is a Kind, because no definition that could be given of the name ‘animal’ could •include the properties common to all animals or •supply premises from which the remainder of those properties could be inferred. A combination of properties that doesn’t give evidence of the existence of any other independent special features doesn’t constitute a Kind. So white horse is not a Kind, because horses that agree in whiteness don’t agree in anything else except •the qualities common to all horses and •whatever causes or effects whiteness has.

On the principle that there should be a name for everything that we have frequent occasion to make assertions about, there obviously ought to be a name for every Kind. It is the very meaning of ‘Kind’ that the individuals composing a Kind have indefinitely many properties in common; from which it follows that the Kind is a subject to which many predicates (if and when we have them) will have to be applied. And so we have the third requirement for a philosophical language: there must be a name for every Kind, i.e. there must not only be a •terminology but also a •nomenclature.

As far as I know, Whewell is the first writer who has regularly given different meanings to the words ‘nomenclature’ and ‘terminology’. But the distinction he has drawn between them is real and important, so his example is likely to be followed by those who come after him; and. . . .a vague sense of the distinction can be seen to have influenced the use of the terms in common practice, before he pointed out the advantages of discriminating them philosophically. When Lavoisier and Guyton-Morveau reformed the language of chemistry, everyone would say, their reform consisted in the introduction of a new •nomenclature, not a new •terminology. ‘Linear’, ‘lanceolate’, ‘oval’, or (descriptions of leaves) ‘oblong’, ‘serrated’, ‘dentate’, and ‘crenate’ belong to the terminology of botany, while the names Viola odorata and Ulex Europaeus belong to its nomenclature.

We can define ‘nomenclature’ thus:

The collection of the names of all the Kinds that any branch of knowledge deals with; or (more properly) of all the lowest Kinds, or infimæ species—the species that can be subdivided but not into Kinds.

These lowest Kinds generally fit with what in natural history are termed simply ‘species’. Science has two splendid examples of a systematic nomenclature; •that of plants and animals, constructed by Linnæus and his successors, and •that of chemistry, which we owe to the illustrious group of chemists who flourished in France toward the close of the eighteenth century. In these two departments, not only does every known ‘species’ or lowest Kind have a name assigned to it, but when new lowest Kinds are discovered names are at once given to them on a uniform principle. In other sciences the nomenclature is not at present constructed on any system, either •because (as in geometry) the species to be named are not numerous enough to require one or •because (as in mineralogy) no-one has yet suggested a suitable principle for such a system—and this lack of a scientifically constructed nomenclature is what is mainly slowing the progress of mineralogy.

§5. A word that carries on its face that it belongs to a nomenclature seems at first sight to differ from other concrete general names in this:

Its meaning doesn’t reside in its •connotation, i.e. in the attributes implied in it, but in its •denotation, i.e. in the particular group of things it is appointed to designate; so it can’t be unfolded by a definition but must be made known in another way.

But this appears to me to be wrong. A word belonging to a nomenclature differs, as I see it, from other words mainly in this:

Besides its ordinary connotation it also has a special one; besides connoting certain attributes, it also connotes that those attributes mark off a Kind.

The term ‘peroxide of iron’ belongs by its form to the systematic nomenclature of chemistry, so it bears on its face that it’s the name of a particular Kind of substance. It also connotes—as does the name of any class—some portion of the properties common to the class; in this instance the property of being a compound of iron and the largest dose of oxygen that iron will combine with. . . . When we say of the substance before us that it is ‘the peroxide of iron’, we are saying (i) that it is a compound of iron and a maximum of oxygen and (ii) that this compound is a particular Kind of substance.

This second part of the connotation of a word belonging to a nomenclature is as essential as the first part, while the definition only declares the first; which is what makes it appear as though the meaning of such terms can’t be conveyed by a definition. But this appearance is fallacious. The name Viola odorata denotes a Kind, a certain number of whose features—sufficient to distinguish it—are stated in botanical works. This list of features is surely, as in other cases, a definition of the name. Some say:

•‘No, it’s not a definition. The name Viola odorata doesn’t mean •those features; it means •that particular group of plants, and the features are selected from a much greater number merely as marks to recognise the group by.’

I reply that the name doesn’t mean that group, because it applies to that group only for a long as the group is believed to be an infima species; if we discovered that several different Kinds have been muddled together under this one name, no-one would go on applying the name Viola odorata to the whole group; if we didn’t just drop it, we would apply it to one only of the Kinds in the group. What is imperative, therefore, is not that the name shall denote one particular collection of objects, but that it shall denote a Kind, a lowest Kind. The form of the name declares that, come what may, it is to denote an infima species; and that therefore the properties it connotes—the one expressed in the definition—are to be connoted by it only for as long as we believe that those properties in combination •indicate a Kind and •aren’t to be found in more than one Kind.

With the addition of this special connotation that is implied in the form of every word that belongs to a systematic nomenclature, the set of features that is used to discriminate each Kind from all other Kinds (and is a real definition) constitutes. . . .the whole meaning of the term. It may be objected:

‘As often happens in natural history, the set of features may be replaced by another set that is better suited for the purpose of distinction, yet the word, still denoting the same group of things, isn’t thought have changed its meaning.’

But this can happen with any other general name: in reforming a name’s connotation we may leave its denotation untouched; and it is generally desirable to do so. This doesn’t stop the connotation from being the real meaning, because wherever the characters set down in the definition are found we immediately apply the name—and anything that exclusively guides us in applying the name must constitute its meaning. If we find that the characters are not exclusive to one species as we had thought it to be, we stop using the term co-extensively with the characters; but that’s because the failure of the other part of the connotation, namely the condition that the class must be a Kind. The connotation, therefore, is still the meaning; the set of descriptive characters is a true definition; and the meaning is unfolded not indeed (as in other cases) by the definition alone, but by the definition and the form of the word taken together. [You may have noticed that Mill’s response to the indented passage above ignores its final clause ‘the word isn’t thought to have changed its meaning’.]

§6. I have now analysed what is implied in the two principal requisites of a philosophical language: •precision, or definiteness [chapters 4 and 5] and •completeness [the present chapter up to here]. Further remarks about how to construct a nomenclature must wait until I come to Classification, because the way of naming the Kinds of things must be subordinate to the way of arranging those Kinds into larger classes. Some of the minor requirements of terminology are well stated and illustrated in Whewell’s ‘Aphorisms regarding the Language of Science’ in his Philosophy of the Inductive Sciences. I shan’t discuss these because they are of secondary importance from the point of view of Logic; and shall discuss only one more quality which, next to the two already treated, appears to be the most valuable the language of science can possess. A general notion of it can be gathered from the following aphorism:

Whenever the nature of the subject permits our reasoning processes to be safely carried on mechanically, the language should be constructed on principles that are as mechanical as possible; and when that is not the case, the language should be constructed in such a way as to place the greatest possible obstacles to a merely mechanical use of it.

I’m aware that this maxim requires much explanation, which I now proceed to give. First, what is meant by ‘use a language mechanically’? We have the complete or extreme case of this when language is used without any consciousness of a meaning, and with only the consciousness of using certain visible or audible marks in conformity with technical rules previously laid down. This extreme case occurs only in the figures of arithmetic, and (still more) in the symbols of algebra—a language unique in its kind, and perhaps coming as close to perfection for its intended purpose as any creation of the human mind. It is perfect because of how completely it is adapted to a purely mechanical use. The symbols are mere counters, without even the semblance of a meaning apart from the convention that is renewed each time they are used, and is altered at each renewal, the same symbol a or x being used on different occasions to represent things that have nothing in common except that they can be numbered. So there’s nothing to distract the mind from the mechanical operations that are to be performed on the symbols, such as squaring both sides of the equation, multiplying or dividing them by the same or by equivalent symbols, and so forth. Each of these operations corresponds to a syllogism—represents one step of a process of reasoning not about the symbols but about the things they signify—but as it has been found possible to create a technical form such that by conforming to it we can reason correctly, we can completely achieve what we’re aiming at without our ever thinking of anything but the symbols. Being intended to work merely as mechanism, these symbols have the qualities that mechanism ought to have: they are of the least possible bulk, so that they take up very little room and waste no time in their manipulation; and they are compact, and fit together so closely that in almost every case the eye can take in all at once the whole operation they are being used to perform.

These admirable properties of the symbolic language of mathematics have led many thinkers to to regard the symbolic language in question as the ideal type [see Glossary] of philosophical language generally; to think that names in general or (as they like to say) signs

are fit for use in thinking in proportion as they can be made to approximate to

the compactness, the entire unmeaningness, and the ability to be used as counters without a thought of what they represent

that are characteristic of the a and b, the x and y, of algebra.

This has led to optimistic views about the acceleration of the progress of science by means which (in my view) •can’t possibly contribute to that end and •is a part of the exaggerated estimate of the influence of signs that has done a lot to prevent the real laws of our intellectual operations from being rightly understood.

For one thing, a set of signs by which we reason without thinking of their meaning can’t be of any use except in our •deductive reasoning. In our direct •inductions we can’t for a moment dispense with a distinct mental image of the phenomena, because the whole operation depends on a perception of the respects in which those phenomena agree and differ. Furthermore, this reasoning by counters is suitable for only a very limited portion even of our deductive processes. In our reasonings about numbers, the only general principles we ever need to introduce are:

•Things that are equal to the same thing are equal to one another, and

•The sums or differences of equal things are equal,

with their various corollaries. No hesitation can ever arise about the applicability of these principles, because they are true of all magnitudes whatever; and, what’s more, every application that can be made of them can be reduced to a technical rule. . . . But when the symbols represent something other than mere numbers—e.g. straight or curved lines—we have to •apply theorems of geometry that aren’t true of all lines and •select the ones that are true of the lines we are reasoning about. And how can we do this unless we keep completely in mind what particular lines these are? Since additional geometrical truths may be introduced into our deductive reasoning at any stage of its progress, we mustn’t allow ourselves, even for an instant, to use the names mechanically. . . .without an image attached to them. It is only after establishing that the solution of a question about lines •can be made to depend on a previous question about numbers—i.e. (in technical terms) •can be reduced to an equation—that the unmeaning signs become available and the nature of the facts we are investigating can be dismissed from the mind. Until the establishment of the equation, the language in which mathematicians do their reasoning doesn’t differ in character from that of close reasoners on any other kind of subject.

I don’t deny that when any correct reasoning is put into the syllogistic shape it is conclusive from the mere form of the expression, provided that none of the terms is ambiguous; and this is one of the facts that have led some writers to think that if all names were constructed and defined so skillfully that there was no ambiguity, this improvement in language would not only •make the conclusions of any deductive science as certain as those of mathematics, but would also •reduce all reasonings to the application of a technical form, and enable their conclusiveness to be rationally assented to after a merely mechanical process, as is undoubtedly the case in algebra. But apart from geometry, the conclusions of which are already as certain and exact as they can be made, the science of number is the only one where the practical validity of a bit of reasoning can be apparent to anyone who looks only at the reasoning itself. Whoever has assented to what I said in Book III about •the Composition of Causes and •the superseding of one set of laws by another is aware that geometry and algebra are the only sciences whose propositions are categorically true; the general propositions of all the other sciences are true only hypothetically, supposing that no counteracting cause happens to interfere. So even a conclusion derived from admitted laws of nature by a formally correct deduction won’t have more than a hypothetical certainty. At every step we must make sure that no other law of nature has superseded, or intermingled its operation with, the laws that are the premises of the reasoning; and how can this be done by merely looking at the words? We must be constantly •thinking of the phenomena themselves and indeed constantly •studying them, learning what the special features are of every case to which we try to apply our general principles.

The algebraic notation, considered as a philosophical language, is perfect for the subjects in which it is commonly used, namely those where the investigations have already been reduced to discovering a relation between numbers. But while it is admirable for that purpose, the properties that make it so do not qualify it as the ideal model of philosophical language in general; indeed, the more nearly the language of any other branch of science approaches to the algebraic notation, the less fit that language is to do its own work. On all other subjects, instead of devices to prevent us from being distracted by thoughts of the meaning of our signs, we ought to want devices to make it impossible that we should ever lose sight of that meaning even for an instant.

With that as our aim we should put as much meaning as possible into the formation of the word itself, making the aids of derivation and analogy available to keep us in mind of all that the word means. In this respect there’s an immense advantage for languages like German that form their compounds and derivatives from native roots rather than from those of a foreign or dead language, as do English, French, and Italian. The best are those that form their compounds according to fixed analogies corresponding to the relations among the ideas to be expressed. All languages do this to some extent, especially—among modern European languages—German; but even that is inferior to Greek, in which the relation between the meaning of •a derivative word and that of •its primitive is in general clearly marked by its how it is formed. . . .

But words, however well constructed originally, are always tending (like coins) to have their inscription worn off by passing from hand to hand; and the only possible way of reviving it is to keep stamping it afresh by habitually thinking about the phenomena themselves and not resting in our familiarity with the words that describe them. [Mill develops this thought, saying that for someone who is ‘content to live among these formulae’ the formulae themselves will gradually lose their meaning. He concludes:] It is as necessary in every non-mathematical subject that the things we are reasoning about should be conceived by us concretely and ‘clothed in details’ as it is in algebra that we should keep all individualising special features carefully out of view.

That is all I shall say here about the Philosophy of Language. [That is the only occurrence of the phrase ‘philosophy of language’ in the entire System of Logic, except in one footnote.]

Chapter 7. Classification, as subsidiary to induction

§1. As I have said before, there’s a kind of classification of things that is inseparable from giving them general names. Every name connoting an attribute automatically divides all things whatever into two classes—those that have the attribute and those that don’t, those that the name can be applied to and those that it can’t. And this divides all things that actually exist, whether known to us or not, and indeed all that can be imagined to exist.

I have nothing more to say about this kind of classification. The classification that needs to be discussed is altogether different; it is a separate act of the mind. In the first kind of classification the arrangement of objects in groups and the distribution of them into compartments is a mere side-effect of the use of names given simply to express some of their qualities. In this second kind of classification the arrangement and distribution are the main object; naming has to be made to •conform itself to this more important operation—it doesn’t •govern it.

This kind of classification is a device for •the best possible ordering of the ideas of objects in our minds; •for causing the ideas to accompany or succeed one another in a way that will give us the greatest command over the knowledge we already have, and lead most directly to the acquisition of more. Given that these are its purposes, the general problem of classification may be stated thus: To bring it about that things are thought of in such groups, and those groups in such an order, as will best favour their laws’ being discovered and remembered.

Unlike the other, this kind of classification refers only to •real objects and has nothing to do with •objects that are merely imaginable. It aims to organise our thinking about things whose properties we are already acquainted with—those things and no others. But it embraces all really existing objects. We can’t constitute any class properly except in reference to a general division of the whole of nature; we can’t determine which group object x can most conveniently be assigned to without taking into consideration all the varieties of existing objects, all at least which have any degree of affinity with x. No one family of plants or animals could have been rationally constituted except as part of a systematic arrangement of all plants or animals; and no such general arrangement could have been properly made without first determining the exact place of plants and animals in a general division of nature.

§2. There is no property of objects that we can’t, if we please, use as the basis for a classification or mental grouping of those objects; and in our first attempts we’re likely to select properties that are simple, easily conceived, and perceptible on a first view that isn’t prepared for by any previous process of thought. Thus Tournefort’s arrangement of plants was based on the shape and divisions of the corolla; and what’s commonly called the Linnæan (though Linnæus also suggested another and more scientific arrangement) was based mainly on the number of the stamens and pistils.

But these classifications, which are initially attractive because they make it easy to discover what class any individual belongs to, are usually not much good for the purpose of Classification that I am now talking about. The Linnæan arrangement is good for getting us to think together of all those kinds of plants that have the same number of stamens and pistils; but thinking of them in that way isn’t much use because we seldom have anything to say about all and only the plants with a given number of stamens and pistils. . . . And as this property is of little importance or interest, remembering it accurately doesn’t matter either. And ·it’s worse than that·: by habitually thinking of plants in •those groups we’re prevented from habitually thinking of them in •groups with more properties in common; so that this classification, if we stick to it, must do harm to our habits of thought.

The purposes of scientific classification are best served when a classification x surpasses any possible rival classification y because of two ways in which

•the general propositions that can be made about x’s groups surpass •the general propositions that can be made about y’s groups,

namely that

•there are more of them,. and

•they are more important.

So objects should be classified (if possible) according to properties that are •causes of many other properties, or at least are sure •marks of them. Causes are preferable, as being the surest and most direct of marks, and also as being themselves the properties that it’s most useful for us to focus on. But the property that is the •cause of the main special features a class is—unfortunately—usually not fitted to serve also as the ·mark or· diagnostic of the class. Instead of the cause, we usually have to select some of its more prominent effects, which may serve as marks of the other effects and of the cause.

A classification formed in this way is strictly scientific or philosophical, and is commonly called a ’natural’ classification or arrangement, as against a ‘technical’ or ‘artificial’ one. The phrase ‘natural classification’ seems most appropriate for arrangements that correspond to the spontaneous tendencies of the mind, placing together in groups the objects most similar in their over-all appearance; in contrast to the ‘technical’ systems which, by grouping things according to their agreement in some arbitrarily selected feature, often put into the same group objects that have no over-all resemblance and put into different and remote groups objects that are very much alike. ·That is a remark about ‘natural classification’ considered intuitively as a phrase in everyday speech; but it also has a bearing on what is scientifically proper·. The claim of a classification to count as ‘scientific’ can be strongly supported by its being a ‘natural classification’ in this ·everyday informal· sense also. Why? Because the test of its scientific character is the number and importance of the properties shared by all objects included in a group; and properties on which the over-all appearance depends are (if only because of that dependence) •important, and in most cases •numerous. But although everyday naturalness is a strong recommendation it isn’t something that a scientific classification has to have: it can happen that the most obvious properties of things may be of trifling importance compared with others that are not obvious. The Linnæan classification has been accused of ‘absurdity’ in grouping the violet with the oak; actually, it doesn’t do that, but it certainly does cut across natural affinities, and groups together things quite as unlike as the oak and the violet are. But the apparently wide difference that makes that grouping such a convincing example of a bad arrangement depends—to the layman’s eye—mainly on mere size and texture; now if we insisted on finding the classification that would involve the least risk of similar groupings, we would return to the obsolete division into •trees, •shrubs, and •herbs. That is of primary importance with regard to over-all appearance, but it corresponds to so few differences in the other properties of plants. . . .that a classification based on it would be as completely artificial and technical as the Linnæan one—quite apart from the unclarity of its lines of demarcation.

Our natural groups, therefore, must be based on the unobvious properties of things when these are of greater importance. But in such cases there must be some other property or set of properties, more easily recognisable by the observer, that co-exist with the properties that are the real basis of the classification and can be used as marks of them. (A natural arrangement of animals must be mainly based on their internal structure, but (as Comte remarks) it would be absurd if we couldn’t determine the genus and species of an animal without first killing it!) So De Blainville’s zoological classification is probably the one we should favour: it is based on the differences in animals’ coverings—skin, shell, rind, etc.—which correspond much more accurately than you might think to the really important differences, both in the other parts of the structure and in the habits and history of the animals.

This shows more strongly than ever how much we have to know about objects’ properties if we are to make a good classification of them. One of the things that make such a classification useful is that by drawing attention to the properties that •it is based on and (if the classification is good) •are marks of many others, it helps us to discover those others; so we see how •our knowledge of things and •our classification of them tend to go on improving each other.

I said just now that the classification of objects should follow those of their properties that indicate not only the most numerous but also the most important special features. What does ‘important’ mean here? It’s a relational term: a given feature is ‘important’ for one purpose, not for others; so the same objects may be classifiable in several different ways, each of them legitimate. Each science or art classifies things according to the properties that fall within its special realm or that it must take account in order to accomplish its special practical end. A farmer doesn’t divide plants (like a botanist) into dicotyledons and monocotyledons, but into useful plants and weeds. A geologist divides fossils not (like a zoologist) into families corresponding to those of living species, but into fossils of the paleozoic, mesozoic, and tertiary periods, above the coal and below the coal, etc. Are whales fish? That depends on why we’re thinking about them. Whewell again:

‘If we are speaking of the internal structure and physiology of the animal, we must not call them “fish”, for in these respects they deviate widely from fishes; they have warm blood, and produce and suckle their young as land quadrupeds do. But this wouldn’t stop us from speaking of the “whale-fishery”, and calling such animals “fish” on all occasions connected with this use; for what matters here is the animal’s living in the water and being caught in a manner similar to other fishes. A plea that human laws that mention fish do not apply to whales would be rejected at once by an intelligent judge.’

These classifications are all good for the purposes of their own particular departments of knowledge or practice. But when we’re studying objects not for any special practical end but for the sake of extending our knowledge of the whole of their properties and relations, we must consider as the most important attributes those that •contribute most, either by themselves or through their effects, to make the things like one another and unlike other things; •that give to the class composed of them the most marked individuality; •which fill (as it were) the largest space in their existence [Mill’s phrase] and would most impress the attention of a spectator who knew all their properties but wasn’t specially interested in any of them. Classes formed on this principle may be called, in a more emphatic way than any others, ‘natural groups’.

§3. On the subject of these groups Whewell lays down a theory, based on an important truth, that he has expressed and illustrated very felicitously in some respects, but with what seems to me to be some admixture of error. Here is what he says:

‘Natural groups are given by Type, not by Definition. . . And this accounts for the indefiniteness and indecision that we frequently find in the descriptions of such groups, and that must appear so strange and inconsistent to anyone who thinks they describe groups that the botanist has arbitrarily chosen. Thus we’re told that in the family of the rose-tree the ovules are very rarely erect, the stigmata are usually simple. You might ask “What’s the use of such loose accounts as these?” The answer is that they aren’t inserted to •distinguish the species but to •describe the family, and the total relations of the ovules and the stigmata of the family are better known by this general statement. . . .

‘These views—of classes determined by characters that can’t be expressed in words—of propositions that state what happens not •in all cases but only •usually—of particulars that are included in a class though they don’t fit the definition of it—may surprise you. They are contrary to many generally accepted opinions about the use of definitions and the nature of scientific propositions, and will probably strike many people as illogical and unphilosophical. But that’s because the mathematical and mathematico-physical sciences have largely determined men’s views of the general nature and form of scientific truth, while Natural History hasn’t yet had time or opportunity to exert its influence on habits of philosophising. Its classifications and definitions are much less indefinite and inconsistent than any others apart from those of mathematics; and the ways in which approximations to exact distinctions and general truths have been made in Natural History may be worth studying, even for the light they throw on the best ways of pursuing truth of all kinds.

‘Though in a natural group of objects a definition can no longer be of any use as a regulative principle, classes aren’t left quite loose, without any certain standard or guide. The class is steadily fixed, though not precisely limited; it is given, though not circumscribed; it is determined, not by a boundary-line around it but by a central point within; not by what it strictly excludes, but by what it eminently includes; by an example, not by a rule; in short, we are directed not by a definition but by a Type. [As Whewell will show shortly, ‘eminently’ here means ‘in a marked and prominent manner’; our topic here is Types, and ‘eminently’ goes with ‘typically’.]

‘A Type is an example of any class (e.g. a species of a genus) that is considered as eminently possessing the character of the class. The species that have a greater affinity with this type-species than with any others jointly form the genus, and are arranged around it, deviating from it in various directions and different degrees. Thus a genus may consist of several species that approach very near the type and have an obvious right to be in the genus, while there may be other species that straggle further from this central knot and yet are clearly more connected with it than with any other. There might be some species whose place is dubious—ones that appear to be equally bound to two generic types—but it’s easy to see that this wouldn’t destroy the reality of the generic groups, any more than the scattered trees of the intervening plain prevent us from speaking intelligibly of the distinct forests of two separate hills.

‘The type-species of every genus. . . ., then, is one that possesses all the characters and properties of the genus in a marked and prominent manner. The type of the Rose family has alternate stipulate leaves, lacks albumen, has non-erect ovules and simple stigmata; and in addition to these features that distinguish it from the exceptions or varieties of its class, it has the features that make it prominent in its class. It is one of those that clearly possess several leading attributes. Thus, even when we can’t say of any one genus that it must be the type of the family, or of any one species that it must be the type of the genus, we still know something about it: the type must be connected by many affinities with most of the others of its group; it must be near the centre of the crowd, not one of the stragglers.’

. . . .Whewell here states one of the principles of a Natural Classification; he does it very clearly and forcibly but without (I think) making all necessary distinctions. What this principle is, what its limits are, and how Whewell seems to me to have overstepped them, will appear after I have laid down another rule of Natural Arrangement, one that appears to me still more fundamental.

§4. You’ll be familiar by now with something that I keep repeating because of the great confusion it is commonly involved in, namely the general truth that there are in nature

distinctions of Kind; not •consisting in a given number of definite properties plus their effects but •running through the whole nature of the things so distinguished—running through their attributes generally.

Our knowledge of the properties of a Kind is never complete. We are always discovering, and expecting to discover, new ones. Where the distinction between two classes of things is not one of Kind, we expect to find their properties alike except where there’s some reason for them to be different. In contrast with that, when the distinction is one of Kind, we expect to find the properties different unless there’s some cause for their being the same. All knowledge of a Kind must be obtained by observation and experiment on the Kind itself. What about inferring some of its properties from the properties of other things that aren’t connected with it by Kind? The most you’ll get from that is the sort of presumption usually called an ‘analogy’, and generally a weak one at that.

The members of a true Kind have countlessly many common properties; there’s no getting to the end of them; so the same is true of the general assertions that can be made about that kind, or that are certain to be made later on as our knowledge extends. ·Now put that beside this·: The very first principle of natural classification is to form the classes so that the objects composing each class may have the greatest number of properties in common. This principle tells us that every natural classification should recognise and adopt into itself every distinction of Kind that exists among the objects it professes to classify. To •pass over any distinctions of Kind, •replacing them by definite distinctions [see Glossary] that don’t point to underlying unknown differences would be to •pass over classes with more attributes in common, •replacing them by classes with fewer; and this would undermine the Natural Method of Classification.

That is why natural arrangements have always conformed to distinctions of Kind so far as these were known at the time; this has been true even when the makers of the arrangement •didn’t have any sense of the distinction of Kinds and •were merely pushing ahead with their own inquiries.

The species of plants are not only real Kinds but are probably all real lowest Kinds, infimæ species [Latin]. We are free to subdivide any of them into smaller sub-classes, but the subdivision would necessarily be founded on definite distinctions, not pointing (apart from what may be known of their causes or effects) to any difference beyond themselves. Because a natural classification is based on real Kinds, its groups are certainly not conventional: they don’t depend on an arbitrary choice by the naturalist. But it doesn’t follow, and I think it isn’t true, that these classes are determined by a •type and not by •characters. To determine them by a type is as sure a way of missing the Kind as if we selected a set of characters arbitrarily. They are determined by characters, but these are not arbitrary. The problem is to find a few definite characters that point to the multitude of indefinite ones. Kinds are classes separated by an impassable barrier; we have to look for marks that will tell us which side of the barrier an object belongs on. The characters that will do this best are the ones we should choose; if they are also important in themselves, so much the better! When we have selected the characters we sort out the objects according to those characters and not, I think, according to their resemblance to a type. We don’t compose the species Ranunculus acris out of all plants that are suitably like a model buttercup, but out of plants with •certain characters selected as marks by which we might recognise the possibility of a common parentage; and the list of •those characters is the definition of the species.

All Kinds are classes, of course, but it is not the case that all the classes in a natural arrangement must be Kinds. There aren’t enough distinctions of Kinds to make up the whole of a classification. Very few of the genera of plants, or even of the families, can be pronounced with certainty to be Kinds. The great plant-distinctions

•Vascular and Cellular,

•Dicotyledonous and Monocotyledonous,

•Exogenous or Endogenous,

are perhaps differences of kind: the lines of demarcation dividing those classes seem (though even on this I’m not sure) to go through the whole nature of the plants. But the different species of a genus, or different genera of a family, usually have only a limited number of characters in common. A Rose doesn’t seem to differ from a Rubus, or the Umbelliferæ from the Ranunculaceæ, in anything much except the characters botanically assigned to those genera or those families. Unenumerated differences certainly do exist in some cases; there are families of plants which have their own special chemical composition, or yield products having special effects on the animal economy. The Cruciferæ and Fungi contain an unusual proportion of nitrogen; the Labiatæ are the chief sources of essential oils; the Solaneæ are very commonly narcotic, etc. In these and similar cases there are possibly distinctions of Kind; but it’s by no means indispensable that there should be. Genera and Families can be eminently natural although they are marked out from one another by a limited number of properties, provided •those properties are important and •the objects contained in each genus or family resemble each other more than they resemble anything outside the genus or family.

After recognising and defining the infimæ species, the next step is to arrange these into larger groups, making the groups correspond to Kinds wherever possible, but in most cases without any such guidance. And in doing this we are usually guided, as is natural and proper, by resemblance to a type. We form our groups around certain selected Kinds, each of which serves as a sort of exemplar of its group. But though the groups are suggested by types, I can’t agree with Whewell

•that when a group has been formed, it is determined by the type; •that in deciding whether a species belongs to the group we refer to the type and not to the characters; •that the characters ‘can’t be expressed in words’ [see here above].

This ·indented· assertion is inconsistent with Whewell’s own statement of the basic principle of classification, namely that ‘general assertions shall be possible’ [quoted from Whewell’s book, but not from the passage Mill quoted a little while back]. If the class didn’t have any characters in common, what general assertions about it could there be? Nothing could be predicated of the class except that they are all more like one another than like anything else. The truth is, on the contrary, that every genus or family is framed [see Glossary] with distinct reference to certain characters, and is composed first and principally of species that agree in having all those characters. To these are added, as a sort of appendix, the few other species that have nearly all the properties selected; some lack one property, some lack another, and while they agree with the rest almost as much as these agree with one another, they don’t as closely resemble any other group. Our conception of the class continues to be based on the characters; and the class might be defined as ‘things that either •have that set of characters or •resemble the things that do so more than they resemble anything else’.

This resemblance isn’t an ultimate unanalysable fact like the resemblance between simple sensations. Even distant resemblances are created by the possession of common characters. Whatever resembles the genus Rose more than it resembles any other genus does so because it possesses more of the characters of that genus than of any other genus. And there can’t be any real difficulty in representing by a list of characters the nature and degree of the resemblance that strictly entitles any object to be included in the class. There are always some properties common to all the included things. There will often be properties that some included things don’t have; but the objects that don’t have one property do have the others; the resemblance that fails in some particulars must be made up for in others. So the class is constituted by the possession of •all the characters that are universal and •most of the others, i.e the ones that allow exceptions. If a plant had erect ovules, divided stigmata, no albumen, and no stipules, it might not be classed among the Rosaceæ. But it could lack any one (or perhaps more than one) of these characteristics and still not be excluded. The purposes of a scientific classification are better served by including it. Because it agrees so nearly, in its known properties, with the sum of the characters of the class, it is likely to resemble that class more than any other in such of its properties as haven’t yet been discovered.

So natural groups are determined by characters just as much as artificial classes are. . . . But what are relevant are not merely •the characters that are rigorously common to all the objects included in the group, but •the entire body of characters such that: most members of the class have all of them, and all members of the class have most of them. That’s why our conception of the class—the image in our minds that represents it—is that of a specimen that has all the characters; most naturally a specimen which, by having them all in the highest degree they are ever found, is the best fitted to show clearly and decisively what they are. It is by a mental reference to this standard that we usually and advantageously determine whether any individual or species belongs to the class; but this standard isn’t a substitute for the definition of the class—it’s an illustration of it. And this, as it seems to me, is all the truth there is in the doctrine of Types. . . .

§5. A Nomenclature in science is (I repeat) a system of the names of Kinds. These names, like other class-names, are defined by the list of characters distinctive of the class. The only merit a set of names can have beyond this is for them to be constructed in such a way as to give as much information as possible: so that a person who knows the thing can get all the help the name can give in remembering what he knows; while someone who doesn’t know it can get—merely by being told its name—as much knowledge about it as the case admits of.

There are two ways of giving the name of a Kind this sort of significance. The better of them is when the word can be made to indicate by its formation the very properties it is designed to connote. The name of a Kind doesn’t (of course) connote all its properties because these are inexhaustible; it connotes enough of its properties to distinguish it—ones that are sure marks of all the rest. Now, it rarely happens that one property, or even two or three, can do this. To distinguish the common daisy from all other species of plants would require a very long list of characters, and a usable name can’t indicate more than a very small number of these by its etymology or mode of construction. So it’s likely that only one ideally perfect Nomenclature is even possible; and it’s one that we fortunately have an approximation to—namely the Nomenclature of elementary Chemistry. The substances, whether simple or compound, that chemistry deals with are Kinds, and therefore the properties distinguishing each of them from the rest are innumerable; but in the case of compound substances there is one property, the chemical composition, that is by itself sufficient to distinguish the Kind, and it is (with certain reservations not yet thoroughly understood) a sure mark of all the other properties of the compound. (As for simple chemical substances: there aren’t enough of them to require a systematic nomenclature.) All that was needed, therefore, was to make the name of every compound declare its chemical composition on the first hearing, i.e. to form the name of the compound in a uniform way from the names of the simple substances that are its elements. This was done skillfully and successfully by the French chemists, though their nomenclature has become inconvenient for naming the very complicated compounds now known to chemists. The only thing their nomenclature didn’t indicate was the exact proportion in which the elements were combined; and the establishment of the atomic theory made it possible to fill this gap.

But where too many characters are needed to pin down the Kind, and where no one of them is so important as to justify its being singled out to be so indicated [the last clause is verbatim from Mill], there’s a subsidiary procedure that can help us out. Though we can’t indicate the distinctive properties of the Kind, we may be able to indicate the Kinds that are most like it by incorporating into its name the name of the proximate [see Glossary] natural group of which it is one of the species. This is the basis for the admirable binary nomenclature of botany and zoology, in which the name of every species consists of •the name of the genus or natural group next above it, with •a word added to distinguish the particular species. The second part of the compound name is •sometimes taken from one of the features in which that species differs from others of the genus;. . . .sometimes from a detail of an historical nature;. . . . and sometimes the word is purely conventional. . . . It doesn’t matter much which of these is used, because the second part—the ‘specific name’—could at most express, independently of convention, no more than a very small portion of the connotation of the term. . . .

This principle of nomenclature helpfully gives to the names of species the greatest amount of independent significance that the circumstances of the case admit of, and further helps us by immensely economising the use of names and saving us from an intolerable burden on the memory. . . .

Chapter 8. Classification by series

§1. Up to here I have considered the principles of scientific classification only in relation to the formation of natural groups; and this has been the stopping-point for Whewell and most of the others who have attempted a theory of natural arrangement. But there’s another part to the theory, concerning the arrangement of the natural groups into a natural series. This is as important as the first part, yet as far as I know it hasn’t been systematically discussed by anyone except Comte.

As I have already said, the purpose of Classification as an instrument for the investigation of nature is to make us group in our minds objects that •have the greatest number of important common properties and therefore •need to be considered together when we are performing inductions. Our ideas of objects are thus ordered in the way that is best for the successful conduct of inductive inquiries generally. But when the purpose is to conduct some particular inductive inquiry, more is required. To be helpful with that, a classification must bring together those objects the simultaneous contemplation of which is likely to throw most light on the particular subject. Because that subject is the laws of some phenomenon or some set of connected phenomena, that phenomenon or set of phenomena must be chosen as the basis for the classification.

If a classification is to help the study of a particular phenomenon it must (i) bring into one class all Kinds of things that exhibit that phenomenon in some form and some degree, and (ii) arrange those Kinds in a series according to the degree in which they exhibit it, from those that exhibit most of it right down to those that exhibit least. The principal example of such a classification, so far, is presented by comparative anatomy and physiology; and that’s where my examples will come from.

§2. Suppose we want to investigate the laws of animal life. After forming the clearest conception of the phenomenon itself that is possible in the present state of our knowledge, we must (i) erect into one great class (that of animals) all the known Kinds of beings where animal life presents itself, in whatever degree and whatever combinations with other properties. Because some of these Kinds manifest the general phenomenon of animal life in a very high degree, and others in a low degree that is barely sufficient for recognition, we must next (ii) arrange the various Kinds in a series, following one another according to the degrees in which they exhibit the phenomenon—starting with man and ending with the most imperfect kinds of zoophytes.

This is merely to say that we should put the instances that the law is to be inductively collected from into the order implied in the fourth method of experimental inquiry discussed in Book III, namely the method of Concomitant Variations. As I said back there, this is often the only method we can use with assurance of a true conclusion in cases where we can’t do much in the way of artificial experiments to separate features that are usually conjoined. The principle of the method is that

facts that increase or diminish together, and disappear together, are either •cause and effect or •effects of a common cause.

When we know that that relation really does hold between the variations, we can confidently assert that the facts themselves are connected, this being either a law of nature or a mere empirical law, according to circumstances.

It is simply obvious that the use of this method must be preceded by the formation of a series such as I have described; and the mere arrangement of a set of objects in a series according to the degrees in which they exhibit some fact whose law we are seeking is too naturally suggested by the needs of our inductive operations to require much illustration here. But in some cases the arrangement required for the •special purpose becomes the determining principle of the classification of the same objects for •general purposes. This will naturally and properly happen when the laws that the •special inquiry aims to discover play such a big part in the •general character and history of the objects whose laws they are—exercise so much influence in determining all the phenomena of which the objects are either the agents or the theatre[1]—that all other differences among the objects should be regarded as mere modifications of the one phenomenon sought, effects caused by the interplay between some incidental circumstance and the laws of that phenomenon. Thus the differences between one class of animals and another may reasonably be considered as mere modifications of the general phenomenon animal life, modifications arising either

•from the different degrees in which that phenomenon is manifested in different animals, or

•from the intermixture of the effects of incidental causes that are special to the nature of each;

in the latter case the effects are produced by the general laws of life, which still exercise a predominant influence over the result. If the picture I have drawn is correct, no inductive inquiry about animals can be successful unless it is subordinate to the great inquiry into the universal laws of animal life; and the classification of animals best suited to that one purpose is the most suitable to all the other purposes of zoological science.

§3. To establish a classification of this sort, or even to understand it after it has been established, we need to be able to recognise the essential similarity among all the items that are classed together, including the lower degrees and obscurer forms of it; i.e. to identify with one another all phenomena that differ only in •degree and in •properties that we suppose to be caused by difference of degree. In order to recognise this identity—i.e. this exact similarity—of quality, we have to have a type-species. We must consider as the type [see Glossary] of the class the one amongst its member Kinds that exhibits in the highest degree the properties that constitute the class, regarding the other varieties as. . . .deviations from it because of their inferior intensity of the characteristic properties. Other things being equal, every phenomenon is best studied where it exists in the greatest intensity. That’s where the effects that depend either on •it or on •its causes will also exist in the highest degree. It’s the only place where those effects can become fully known to us, enabling us to learn to recognise their lower degrees, or even their mere rudiments, in cases where direct study of them would have been difficult or impossible. Not to mention that the phenomenon in its higher degrees may have effects or collateral details that don’t occur at all in its lower degrees. . . . Consider for example man, the species in which the phenomena animal and organic life exist in the highest degree. Many subordinate phenomena develop in the course of man’s animated existence but don’t appear at all in the lower animals; yet knowledge of these properties may greatly help us to discover the conditions and laws of the general phenomenon of life that man shares with the lower animals. . . .

§4. The remaining task in this Book is to consider what internal distribution of the series is best—i.e. how best to divide it into Orders, Families, and Genera.

The main principle of division must of course be natural affinity; (a) the classes formed must be natural groups; and I have already said enough about how these are to be formed. But the principles of natural •grouping must be applied in subordination to the principle of a natural •series: the groups must not be constituted in a way that places together in one group things that ought to be at different points on the general scale. The precaution we have to take is this:

(b) The primary divisions must be based not on •all distinctions indiscriminately but on •those that correspond to variations in the degree of the main phenomenon.

The animated nature series should be broken into parts at the points where the variation in the degree of intensity of the main phenomenon (as marked by its principal characters, Sensation, Thought, Voluntary Motion, etc.) starts to show up in conspicuous changes in the various properties of the animal. Such well-marked changes occur where the class Mammalia ends; at the points where Fishes are separated from Insects, Insects from Mollusca, etc. When the primary natural groups are formed in that way they’ll compose the series by mere juxtaposition, without redistribution; each of them corresponding to a definite portion of the scale. Similarly, each family should, if possible, be subdivided in such a way that one part of it stands higher and the other lower, though of course contiguous, in the general scale; and only when this is impossible is it allowable to base the remaining subdivisions on characters that have no determinable connection with the main phenomenon.

Where the principal phenomenon is vastly more important than all other properties a classification could be based on—as it is in the case of animated existence—any considerable deviation from rule (b) is in general sufficiently guarded against by the principle (a) that the groups are to be ·natural, i.e.· formed according to the most important characters. All attempts at a scientific classification of animals, since their anatomy and physiology were first successfully studied, have been made with a certain degree of instinctive reference to a natural series, and have agreed more than they have disagreed with (b) the classification that would most naturally have been based on such a series. But the agreement hasn’t always been complete; and there’s still discussion about which of several classifications best fits the true scale of intensity of the main phenomenon. [Mill gives the example of a classification based on how animals get their nourishment, and agrees with the criticism that] carnivorous and herbivorous animals are found at almost every degree in the scale of animal perfection. . . .

§5. Animals constitute the only large part of nature that it has been found practically possible to classify in conformity with principles (a) and (b). In the case of vegetables [= plants generally] the natural arrangement hasn’t been taken beyond (a) the formation of natural groups. Naturalists haven’t been—and probably won’t ever be—able to (b) form those groups into any series whose terms correspond to real gradations in the phenomenon of vegetative or organic life. Such a difference of degree can be traced between •vascular plants and •cellular plants; the latter class includes lichens, algæ, and other substances whose organisation is simpler and more rudimentary than that of the higher order of vegetables, which brings them nearer to mere inorganic nature. But when we rise much above this point, we don’t find any big differences in the degree to which different plants have the properties of organisation and life. The dicotyledons have a more complex structure and somewhat more perfect organisation than the monocotyledons; and some dicotyledonous families, such as the Compositæ, are rather more complex in their organisation than the rest. But the differences are not conspicuous and don’t promise to throw any particular light on the conditions and laws of vegetable life and development. If they did, the classification of vegetables would have to be made, like that of animals, with reference to (b) the scale or series indicated.

The scientific arrangements of organic nature provide us with our only complete example, so far, of the true principles of rational classification in the formation (a) of groups and (b) of series; but those principles are applicable to all cases where mankind are called on to get a mentally orderly grasp of any extensive subject. When our concern is with art or business, the principles are as much to the point when we are doing science. The proper arrangement of a code of laws, for example, depends on the same scientific conditions [Mill’s phrase] as the classifications in natural history; and there couldn’t be a better preparation for that important ·legal· task than the study of the principles of a natural arrangement not only •in the abstract but also •in their actual application to the class of phenomena for which they were first elaborated, and which are still the best school for learning their use. Bentham was perfectly aware of this the great authority on classification; and his early Fragment on Government—an admirable introduction to a series of writings unequalled in their department—contains clear, sound views (as far as they go) on the meaning of a natural arrangement. Such views could scarcely have occurred to anyone who lived before the time of Linnæus.

NOTES

[1] [This phrase refers to two ways in which an object may be crucial to a sequence of events: •as the cause of the events or •as the stage on which the events run their course.]