Again if I say " A, B, and C are a, and D is different from A, B, and C, therefore D is a" — is not this a bad argument, so glaringly bad that no child and no beast could be got to use it?
But if we amend this semblance of reasoning, and bring it to the form of a real inference — if we say " A, B, and C are a,, and therefore D, which resembles them, is a" we are no longer arguing from mere particulars. We are arguing from the resemblance, from a point or points which D has in common with A, B, and C. It is not because A, B, and C are a, but it is because in them some element $ is a, and because again we find /? in D, that we argue " therefore D is a." For whenever we reason from resemblance we reason from identity, from that which is the same in several particulars and is itself not a particular. And is it not obvious that, in arguing from par ticular cases, we leave out some of the differences, and that we could not argue if we did not leave them out? Is it not then palpable that, when the differences are disregarded, the residue is an universal? Is it not once more clear that, in vicious inferences by analogy, the fault can be found in a wrong generalization?
§ 13. I will conclude with an appeal to common experience. We all know very well that in our daily life we reason habitu ally from the results of past experience, although we may be wholly unable to give one single particular fact in support of our conclusion. We know again that there are persons, whose memory is so good that they recall past details in a way which to us is quite impossible, and who yet can not draw the conclusions which we draw, since they have never gone beyond the reproduction of these details. It is not the collection of particular facts, it is the general impression one gets from these facts which is really the sine qua non of reasoning; and it is that from which we really go to our result.
If you begin the discussion of a question, such as this, with a vicious disjunction, you can not go right. As a preliminary to discussion you have excluded the truth. From the alterna tive — either an explicit syllogism or an inference from particu lars to particulars — you can hardly fail to get a false result. You may infer — The syllogism in extension is no argument, and therefore we go from particulars to particulars. You may CHAP. II THE ARGUMENT FROM PARTICULARS 353 infer — It is not possible to argue from particulars, and there fore we reason always in syllogisms, explicit and (if you like) also extensional. But to me it is nothing which conclusion you adopt. For both are errors, and both at bottom are one and the same error. They are twin branches from one root of inveterate prejudice and false assumption.
§ 14. The present chapter has been so short that I take this opportunity to deliver my mind from a weight that oppresses it. I intend to be guilty of what some readers may think an unpardonable omission. It is true that I do not undertake to criticize every theory from which I dissent; but there is one of those theories which I propose to pass over, that may seem to call for recognition and enquiry. Mr. Spencer, in his Psychology, has developed a view of the nature of inference, which, despite its ingenuity, despite its perception of some of those truths which the syllogism has forgotten, I am obliged to consider fundamentally mistaken. It has always seemed to me so arbitrary and so forced, so far away in the end from the real facts, that I can not believe a discussion of it here would tend to throw any light on the problems of logic.3 More than once, I admit, Mr. Spencer's position in English philosophy induced me to think that I had no right to omit all notice of his peculiar views. The sacrifice of space, the chance that I had failed to follow the process which had brought him his results, did not weigh against the danger that I might have seemed to avoid confronting my own doctrines with those of an established master in the subject. But there came to my mind another consideration, which decided the result and fixed my purpose to omit the examination, late Mr Mill and Professor Bain have both written systematic treatises on logic. They have entertained a view of Mr. Spen cer's powers and philosophical performances which is not mine. Mr. Mill especially has expressed his conviction in such terms, that beside it those praises, I should otherwise have felt were due to Mr. Spencer, would sound like detrac too* Both must have been aware that Mr. Spencer has ^more than once published what appears to be a novel theory of reasomng And yet neither (so far as I know) has examined the 1 peculiar and salient assertions of that theory.
354 THE PRINCIPLES OF LOGIC BOOK II. PT. II And I thought that I might venture on a humble imitation of their common silence. Did they fail to follow Mr. Spencer's demonstrations, did they even think them an unprofitable sub ject, in either case I claim the protection of their authority. But, if neither is the truth and they considered Mr. Spencer to be of one mind with themselves, and to say the same thing in a different form, then once again they unite in excusing me. I surely am not wrong if I too omit all criticism, or at least delay it till I have seen some cause to think that it is wanted.
ADDITIONAL NOTES 1 On the argument from particulars, beside the references given, cf. also Bk. III. I. VII. § 8 and III. II. I. § 5.
2 " The ambiguities inherent in that chapter." These are such that apparently (given sufficient good will) the chapter can mean that we never do or can argue directly from particulars (see Appearance, p. 596, note). With regard to J. S. Mill the questions to be answered, if any one thinks them worth answering, are these, (i) Had J. S. Mill any new view to offer? (ii) If so, what was it? (iii) What is the view logically required by his general position? But I must be forgiven if I go on to add " Let the dead bury their dead."
3 With regard to Mr. Spencer's view I would suggest, as a possi bility, that it never was taken from the facts, but was a development of or from something about Comparison which he found in Hamilton. Reading so few books, Mr. Spencer was naturally more at the mercy of those that he did read.
CHAPTER III THE INDUCTIVE METHODS OF PROOF § I. We have seen that in reality there is no such thing as an inference from the particular to a fresh particular In this chapter we approach a cognate superstition. In England at least if we go with the fashion,1 we all have to believe in an Inductive Logic, which, starting from particular given facts goes on to prove universal truths. Its processes, exact as the strictest syllogism, surrender themselves to the direction of Canons, reputed no less severe than Barbara and believed with reason to be far more fertile. I am afraid I may lose the reader's sympathy when I advise him to doubt the union of these qualities.
§ 2. To question the existence or deny the efficacy of those methods of reasoning (whatever they may be), by which modern science has made its conquests, would of course be absurd. To succeed on a great scale is to prove one's title. And it is not within the scope of this work to investigate either the nature of the processes which science employs, or the amount of evidence which it accepts as proof. What I wish to assert is that, starting from particular perceptions of sense, there is no way of going to universal truths by a process of demonstration perfectly exact, and in all its steps theoretically accurate. The induction of logicians, so far as it professes to make that attempt, I shall try to show will not stand criticism.
§ 3. We need not discuss at any great length the Method which is called Complete Induction.2 To examine a number of individuals and to say of all what you say of each, is in the first place no inference to an universal truth. A collective term, if taken collectively, is no more universal than if taken distributively (p. 82); and the inference, if admitted, does not reach the conclusion which we have in view. But in the second place, the inference itself is inadmissible. In other words, if you start from each and end with each, there is no 356 THE PRINCIPLES OF LOGIC BOOK II. Px. II process; but if you predicate of the collection what is true of each member, there is palpable error. The Induction by way of Complete Enumeration must be rejected as either tautolo- gous or false (cf. Book II. I. Chap. II. § 5).
Or again if we take the Induction in another sense, it changes its character. If first by counting you arrive at all, and then from all pass on to any, that is not a process which need be false or need merely repeat the fact it began with; but then it is not based simply upon the particular data. If a flock of sheep have all had medicine, I know that, within the given enclosure, any sheep has been dosed, and I connect the attributes without thinking of the individuals. The conclusion is valid and is really universal; but it implies a process which goes beyond counting. " This sheep and that sheep and the others are dosed; " that is the first premise; but a second is wanted. We may write it " This sheep and that sheep and the others are every sheep that is within this fold," or again " The fold does not contain any sheep but these which we have counted." It is on the strength of this premise that we go on to conclude, "If any sheep is now within the fold he must have been dosed." We seem to argue from " all " to " any," but the " all " has ceased to be the mere collection.
We have first the assurance that the whole field has been surveyed, and that we have not neglected any relevant matter. Counting is the way in which we attempt to obtain this assur ance. But the enumeration, if it is to be complete, must be qualified by the privative judgment, Nothing in this fold can have been uncounted. The collection is thus identified with every possible sheep that comes under the condition of being in the fold. This is one side of our process. The other side consists in an act of abstraction, and in the selective perception of one connection of attributes throughout our whole subject matter. Then, given an individual possessing the condition of belonging to our fold, we pass at once to the other connected attribute.
Now the procedure by which we get this general connection is in a sense " inductive "; and assuredly once more it has employed counting. But then the counting by itself is not the induction, and is not by itself a generalization. The discrimi native analysis, that goes with the counting, is the real agent CHAP. Ill THE INDUCTIVE METHODS OF PROOF 357 which procures the universal, and which contains the " induc tion " (cf. Book III.). It is this which generalizes from the facts. But it does not go beyond one single case,3 since its validity depends on the privative judgment by which any folded sheep must be one case with the sheep observed.
To repeat, if you confine yourself to mere counting, you get no general result. If you attempt to advance from the basis of mere counting your ground is unsafe. If you proceed from a complete Enumeration, then the warrant of complete ness falls outside the counting. What generalizes is the selec tive perception which isolates and secures the connection of adjectives. But the conclusion depen'ds on the guarantee of completeness. It is valid because the connection is found in a whole, which is warranted to anticipate every possible case of a certain kind.
§ 4. But induction by way of Enumeration is not the method we are asked to believe in.* In the treatise which, partly from merits of its own and partly also from other causes, has threatened to fasten itself on us as a text book, we find the so-called Canons of Induction, collected and de veloped from other writers, and formulated with a show of rigorous accuracy. It is the illusory nature of these self- styled proofs that I wish to point out in the present chapter. We must not be afraid of the shadow of authority. The balance of authority among modern logicians is, I think, against the claim of the inductive proofs, and is not on their side. And perhaps already, from experience we have had, we may be prepared to find that Mr. Mill may at times be mistaken. § 5. We must remember above all things throughout this discussion that the question is not, Can discoveries be made by the use of the Methods? They may be as efficacious in actual practice as is asserted by some, or as practically inadequate and unsuited for work as is affirmed by others. That is not the issue which we have before us. The question we have to answer here is, Are they valid ways of proof, by which we can go from facts to universals?
For that is the claim which the Canons set up. " The * The reader of Mill's Logic will remember, on the other hand, that with him the whole inductive process is taken to stand or fall with a proof by way of incomplete Enumeration.
THE PRINCIPLES OF LOGIC BOOK II. PT. II business of Inductive Logic is to provide rules and models (such as the Syllogism and its rules are for ratiocination) to which if inductive arguments conform, those arguments are conclusive, and not otherwise. This is what the Four Methods profess to be " (J. S. Mill, Logic, Bk. III. ix. § 6). " In saying that no discoveries were ever made by the Four Methods, he affirms that none were ever made by observation and experi ment; for assuredly if any were, it was by processes reducible to one or other of those methods" (ibid.). "But induction is not a mere mode of investigation." " 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 " (Logic, III. ii. §5). We can have now no doubt about the nature of this claim; and this claim it is that we are going to discuss.
§ 6. I shall endeavour to show three things: first that the Four Inductive Methods can not be used if we start with mere facts, that the Canons presuppose universal truths as the material upon which the work is to be done; and that therefore, if valid, the Methods are not inductive at all, in the sense of generalizing from particulars. In the next place I shall briefly exhibit the real nature of the reasoning used in the above Four Methods, and shall point out that its essence is not thus in ductive. And finally I shall show that not one of the Canons is a test of proof, and that by every one you can bring out what is false. None of these three positions depends on the others. If the Canons are invalid, if their essence is not inductive, or if they can not be applied to individual facts — if, in short, any one of these contentions is established, the inductive logic is certainly refuted. And I hope to establish firmly all three. §7. (I.) In the first place there is no doubt at all that the basis, from which we are to start in induction, consists primarily of particular given facts. I need cite no passages to establish this point. We naturally expect then to see on the one side the material as yet untouched by the Methods, and on the other the operation of these agents on the crude subject matter with which they must begin. This natural expectation is doomed to disappointment.
(a) A suspicion of the shock which we are destined to CHAP. Ill THE INDUCTIVE METHODS OF PROOF 359 receive may have come from the effrontery of the Method called " Residues." This estimable exemplar of " our great mental operation " comes up to us placarded as one of " the means which mankind possess for exploring the laws of nature by specific observation and experience," and then openly avows that it depends entirely on " previous inductions." Unless supplied beforehand, that is, with one or more ready-made universal propositions, it candidly declines to work at all. We enquire of " Residues " where we are then to begin, and it says, " I do not know; you had better ask ' Difference/ " We anxiously turn to consider " Difference," and are staggered at once by the distressing extent of the family likeness. A chilling idea now steals into the mind; but we have gone too far to retreat at once, so, resolutely turning our back upon " Residues," we begin our examination.
(b) We look at the samples of the work produced, and we find the same thing turning up everywhere. The material supplied to be dealt with by the Methods is never facts but is always universals. Sometimes an open and professed generalization is used as a starting point. But, where this is not done, the material is never a particular fact. It has always been subjected to such previous operation that it is able at once to be taken and used as a " case " or " instance." But this means that already it is an abstract statement, ideal and not real, capable of repetition with other environment, and without doubt universal. Take the very first instance: " Let the antecedent A be the contact of an alkaline substance and an oil. This combination being tried under several varieties of circumstances, resembling each other in nothing else, the results agree in the production of a greasy and detersive or saponaceous substance " (Logic, III. viii. § i). And this is the raw material which is supplied. Before I begin my induction I am to know already that, under certain sets of definite con ditions exactly known, certain results have followed. But, i1 I know this, I also know that these results will always follow given the conditions. Every one of the instances is already an universal proposition; and it is not a particular fad phenomenon at all.*..
§ 8. It seems at first a strange obliquity of instinct * Cf. Whewell, Philosophy of Discovery, p. 263.
360 THE PRINCIPLES OF LOGIC BOOK II. PT. II choose illustrations which can not illustrate.* But on turning to examine the Canons themselves, our surprise gives place to another feeling. The illustrations have been selected, not according to choice, but from hard necessity. For the Canons are such that ex hypothesi they can not possibly work upon any material but universal propositions.
FIRST CANON.
// two or more instances of the phenomenon under investi gation have only one circumstance in common, the circum stance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.
SECOND CANON.
// an instance in which the phenomenon under investiga tion occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.
THIRD CANON.
THIRD CANON.
// two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phe nomenon.
FOURTH CANON.
Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.
FIFTH CANON.
Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected * There is an exception which I will deal with in § 9 CHAP. Ill THE INDUCTIVE METHODS OF PROOF 361 with it through some fact of causation. (Mill, Logic III viii.)
Consider the phrases "only one circumstance in common" every circumstance in common but one," "nothing in com mon save the absence of that circumstance." Only think for a moment and realize what they mean, and then take on the other hand a given fact of perception. The fact is made a particular fact by the presence of that, the absence of which is postulated beforehand by these formulas. A universal judg ment is made universal by just those attributes which are pronounced indispensable in the material for these Methods. The moment you have reduced your particular fact to a per fectly definite set of elements, existing in relations which are accurately known, there you have left the fact behind you. You have already a judgment universal in the same sense in which the result of your " induction " is universal. Let us take once again the very first instance. The universal which you come to is " that the combination of an oil and an alkali causes the production of soap." The universals which you start with are that an oil and an alkali, if combined under con ditions be and de, in each case produce soap. But how can you deny that these latter are universals? No doubt they are impure; but the result of the " induction " is surely not quite pure. And is an impure universal no universal at all? If you assert this, you deny the efficacy of your " induction." If you will not assert it, then you admit that your "induc tions " are not inductive, since the base they start from is not individual facts. If we regard the formulas for a little steadily, we must surely see that an " instance " which is capable of being so formulated, has had already done upon it that work which we heard the Methods, and the Methods alone, were capable of performing. And, if so, these Methods must retire from the field or withdraw their claims. Some thing like a farce has been played before us, whether we consider the airs and pretences of the Canons, or remember the promises and the boasts of their patron.
§ 9. But I may be reminded of and in fairness I must quote an instance, selected by the author himself, to show that his Methods can deal with common material. And the instance 362 THE PRINCIPLES OF LOGIC BOOK II. PT. II has the greater relevancy here, since he devised it expressly to meet the objection that the conditions of his formulas could not be found in facts.
"If it had been my object to justify the processes them selves as means of investigation, there would have been no need to look far off, or make use of recondite or complicated instances. As a specimen of a truth ascertained by the Method of Agreement, I might have chosen the proposition ' Dogs bark/ This dog, and that dog, and the other dog, answer to ABC, ADE, AFG. 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 burnt; this is BC; I touch it, and am burnt; this is ABC, aBC." (Logic, III. ix. 6.)
The Canons we think are not hard to content if this will satisfy them. But surely their author had forgotten them for the moment. By seeing three barking dogs I perceive that they " have only one circumstance in common." By standing in front of a burning fireplace, and then touching the fire and being burnt, I am to know that the two facts " have every circumstance in common but one" Is not this preposterous? Surely it is clear in the first case that Mr. Mill's way of arguing might prove just as well that all dogs have the mange, and in the second that every fireplace blisters. And these con clusions hardly seem to be sound.* If we have succeeded so far in establishing this point, then the Methods of induction are placed in this dilemma. Because they presuppose universal truths, therefore they are not the only way of proving them. But if they are the only way of proving them, then every universal truth is unproved.
§ 10. (II.) The second assertion I have now to make good, is that the process of the Methods is not inductive. I do not mean merely that, as we have seen, they can not be applied except to universals. I mean in addition that it is not at all * As a test of the writer's accuracy in small points, we may notice that in the second example there is a mistake in the working of the Method. The right conclusion is " Touching burns "; for the fire is not the differential condition. It was there before I touched it, and if it was not there, then we have two differences and another kind of mistake.
CHAP. Ill THE INDUCTIVE METHODS OF PROOF 363 of the essence of their process to bring out a conclusion more general than the premises. The process is one of elimination (cf. Book III. p. 412). By removing one part of an ideal con struction you establish the remainder. And hence the result will be more abstract than the whole original datum, but it need not be more abstract than some of the premises; on the contrary it may be less so.4 If five plums, two apples, and ten nuts balance the scales against three pears, two peaches, and six grapes, when I know that the nuts weigh the same as the grapes, and the apples as the peaches, I infer that the plums and the pears are equal by an ideal process of removing the rest. But if this is " induction," then " x r-f-' 5 — 3 = a + 4 — 2, and therefore x = a" and again " A is either b or c, A is not c, and therefore it is b" will also be inductions. And if everything is induction which is not syllogism, then cer tainly these inferences are all inductive. But such an assump tion would surely be quite erroneous. It finds its parallel in the counterpart mistake, that, because the Inductive Methods are not really " inductive," therefore they are syllogistic.
The Methods are all of them Methods of Residues or Methods of Difference, and they all go to their conclusion in the self-same way. They fix a relation between certain wholes, and then, by the removal of parts of each, establish this relation between the remaining elements. In the Methods of Agreement and Concomitant Variations the principle is the same as it is in the rest. In the former the data are ABC- def, AGH — dij, AKL—dmn. It is then assumed that the d in def, dij, and dmn, can not be produced by a different cause; and hence, since BC, GH, KL are different, they do not produce d. A is the residue or difference, and therefore A is the cause. The process we shall see is vicious, but, such as it is, it is elimination. In Concomitant Variations we seem to have AXBC — d1ef; and then, when A1 becomes A2, we have A2BC — d2ef. From this whole take away XBC - - *ef, 2BC — *ef and the conclusion is A — d. The principle involved is the same throughout, and the apparent failure to see this, and the setting down of two or three co-ordinate axioms for the different Methods, is another sign that the writer had never got really inside his subject. The different Methods are different applications of one single process, and since the premises 364 THE PRINCIPLES OF LOGIC BOOK II. Px. II eliminated may be just as abstract as the conclusion left be hind, this process can hardly be called " inductive."
§ ii. Having seen first of all that the Canons will not work unless applied to universals; having seen, in the second place, that within these limits their procedure is not essentially one of generalization, we come now to the third of our objec tions. The Methods are vicious and the Canons are false.
(III.) I do not mean to say that, for all the purposes of discovery, the flaws in the Methods amount to serious mis takes. Such a contention would lie beyond the scope of my volume. It is certain, however, that independent logicians, such as Dr. Whewell and Professor Jevons in our own coun try, and Professors Lotze and Sigwart in Germany, have taken a view of the process of scientific discovery which is not favourable to the claims of the Four Methods. But whatever may be the usefulness of these Methods, the point here at issue is their validity as proofs.
What I wish to show is that they will not prove anything beyond this or that individual case. They pass to their more general conclusion by illegitimate assumptions.
§ 12. I think the reader will agree that, if a method will prove a false conclusion from premises which are true, then that method must be logically vicious, and its Canon, which serves as a test, must be false. Now it is stated by Mr. Mill himself that the Method of Agreement will prove false con clusions (Logic, Chap. X.). The Method is "uncertain" and has an " imperfection." But it still continues to figure as a proof, and the Canon is left standing in its naked falsity. We also have " axioms " implied in this Method, which can hardly be true if the Method is false, and which yet are left exposed to the daylight. We are told (Chap. X. § i) that in chapters preceding false assumptions have been made, and yet the chapters with all their contents are recommended to us still as a sort of Gospel. And here I must frankly confess myself at a loss. Can the writer really have known that all his Canons were false statements? Whether he did or did not, I will not here enquire, for the discussion would not be likely to profit us. It will be perhaps convenient for the sake of argu ment to assume that he did not know the full vice of all his Methods.
CHAP. Ill THE INDUCTIVE METHODS OF PROOF 365