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 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.
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.
// 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.
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 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 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.
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 ABCdef, 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 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.
The Method of Agreement starts from the premises ABC — def* AGH — dijt AKL — dmn: and its conclusion is that A is the cause of d. The principle it goes on is (as we saw before) that whatever is different in the different cases can be eliminated. And this principle is false, since a consequence, such as d, need not always follow from the same antecedent.5 The generalization is therefore vicious, and the Canon which regulates it is false. The axioms also, given in § 2 of the same eighth chapter, are no less false. To make them true you must qualify them by adding " in this one case." But that means you must destroy their generalizing power.
§ 13. The Method of Difference is no less vicious. f From the premises ABC — def, BC — ef, it goes to the conclusion that A is the cause or an indispensable part of the cause of d. But this conclusion is fatally unsound. A may be here a single factor in the production of d, the presence of which is quite accidental. The rule may be for d to be produced entirely without A, and for A to be present without producing d. The foundation of the Method $ " that whatever can not be eliminated, is connected with the phenomenon by a law " is quite false, unless we add to it " in this one case" and thereby make it ineffectual for the purpose of generalizing.
The Method of Joint Agreement and Difference is essen tially the same, and presents the same flaw. Its premises con sist of ABC — def, AGtt — dij, AKL — dmn, BC — ef, GH — ij, KL — mn. It infers from these the conclusion A — d. The 'mistake is the same as that which vitiated Difference. The right conclusion is that, in these three cases, A has gone to produce d..
In the Method of Residues the process is the same, and is bad for the same reason. From ABC --def, B — f, C — e, the Method goes on at once to A — d. But it could do so legitimately, only if it excluded the possibility of B or C, o *I have of course altered Mill's lettering. If his letters mean any thing, they involve a flagrant petitio; and if they do not, their sug gestion must tend to confuse us.
t For further explanation see Bk. III. II. Chap. III. §§ 11 foil. ^ t There is no material difference between this and what is wrongly given, in the same §3, as different, and as the ground of the Method of Agreement; for you have postulated a connection » your premises. I have given above the real ground of the Method of Agreement.
both, having influenced, and been influenced by, A. Other wise the conclusion like all the rest is vicious, and its Canon is false, unless qualified by the words " in this one case" We come in the end to Concomitant Variations, and the principle of this has, I think, not been formulated with the desirable exactness. In the first place the words whenever in the Canon itself and invariably in the Axiom assigned to it are both ambiguous. If they mean that the groups of elements are causally connected, then this must rest upon a previous Method, and not upon mere facts. And in the second place, if we consider the process as a conclusion from these idealized premises, still it is impossible even then to demonstrate a result which will hold beyond this or that case (or cases). The premises appear to be A^C — fref, A2BC — d2ef, A3BC — dzef, and the conclusion arrived at seems to be A — d. We have apparently to eliminate everything but A — d, which is hence left as proved. But since once again the factors are not isolated, we have the old mistake of Difference once more. The real conclusion is "In this one case (or set of cases) without A no d." Because the modification of A has altered the result, therefore A is relevant to d in this alteration, or series of alterations. I may add that no amount of instances and of " approximation " will suffice to demonstrate logically.
Should however finally the premises not have been so idealized as to be reducible to the formula we have given — if we really have nothing whatever to start with but a certain number of observed concomitances — then there literally is no conclusion at all, for the co-existence always may be mere chance coincidence. And, according as we understand the Canon and the Axiom, we must pronounce them to be either insufficient or false.
§ 14. I have shown that, if used in order to generalize beyond this or that individual instance as prepared for treatment, the Methods are vicious, and their Canons false. Their eliminative process will only show that the whole antecedent has been concerned in producing the whole con sequent (cf. Book III.). The attempt to go further and, by isolating the factors, to transcend the limits of the premises supplied, we have seen has broken down at all points.6 In the premises ABC — def, BC — ef, you are supposed to know that def is connected with ABC, and ef with BC: what you do not yet know is if, in ABC, A is really a factor. For it might be irrelevant, and BC without it might produce def. But now, having BC — ef, and resting on the assumption which we call the Principle of Identity (Book I. Chap. V.), you are sure that, if BC — ef is once true, it will be true for ever. And you proceed from this to argue that BC — def must be false. For to produce def B must have been altered: and since in ABC — def the result is produced with no possible alteration except mere A, A there must be relevant to the presence of def. Hence A in this case (of ABC — def) must be, directly or indirectly, relevant to d. But you must not go further, and try in any way to specify the connection. For you can not do that without closing possibilities, and assum ing something not given in your premises.* And we must not forget that even this conclusion depends on our having assumed in the premises that, in ABC — deff d is not irrelevant. Unless we are perfectly sure beforehand that the whole def has been produced by ABC, we can not advance one single step. This shows once more how absurd it is to imagine that the Methods can be applied to particular facts. They depend entirely on such an artificial preparation of the material supplied, as has already reduced it to the form of an universal. It would be waste of time to dwell further on the detail of the Four (or Five) Methods, since the process in all is the same at bottom.f § 15. We have seen that the Methods are not " inductive," since they will not generalize beyond the given instance. They fail again of being " inductive," since they can not be applied to simple facts. They will not work unless they are supplied with universals. They presuppose in short as their own con dition the result they profess alone to produce. Once more, the essence of their procedure is as much deductive as it is " inductive." The conclusion in some cases has less generality than some of the premises.
On any one of these grounds (and I hope on all of them) *I should like here, and on the whole subject, to refer to Lotze's Logik, II. VII.
f I must refer to the following Book for an account of inference by way of Elimination.
we may set down the Inductive Logic as a fiasco. And, if I am told that these flaws, or most of them, are already ad mitted by Inductive Logicians, I will not retract the word I have used. But to satisfy the objector I will give way so far as to write for fiasco, confessed fiasco.
§ 1 6. If it really is the case that the Methods are not sound; if it really is the case that the Canons are not true; if it really is the case that " induction " is not proof, and that he has all along known this, and been well aware of it — in that case I would suggest to the Inductive Logician that he has provoked a possible harsh remark. And however mistaken that harsh judgment might be, yet I can not help thinking that it would be better if he were to tell the public, what they certainly do not know, and the opposite of which his too large professions have led them to believe. But if, as I suppose, the Inductive Logician himself makes the mistake which his public has accepted — if, that is, while admitting that, like all things human, his Methods have " imperfections," he has no idea that, taken as proofs, they are radically vicious — in that case I will end by expressing the hope of a final agreement.7 By abridging claims that will not stand criticism, and by reforming the root and principle of his fabric, he will bring no ruin to the bulk of his edifice. Even if we confined ourselves to Mr. Mill's Logic, we should find that, when his so-called Four Inductive Methods were wholly removed, and his inference from mere particulars banished as a misunder standing, the more valuable and even the larger part of his discussions on Science would remain untouched.
1 " If we go with the fashion." I have to remind the reader once more that this refers to the year 1883.
2 This account of Complete Enumeration and the Collective Judg ment is very seriously wrong. Indeed what is said in this volume about the Collective Judgment (see Index) needs correction perhaps throughout. For a true account of the matter I must refer the reader to Bosanquet, K & R, pp. 76 foil., and Logic, I, 152 foil. The main point is this, that all counting presupposes and depends on a qualitative Whole, and that the Collective Judgment asserts a generic connection within its group. Hence no mere particulars can be counted. I regret the superficiality of my treatment in this work.
3 " One single case." If this means " One single sheep," it is ob viously wrong; and it is still wrong even if it means "each single sheep." What is true is that the group is taken as a region within which a universal connection holds throughout. Hence, and hence alone, we can use such expressions as " any " and " one case with."
A minor point is that for " any folded sheep " we should read "any sheep folded here." This difference points to the weakness of the Collective Judgment. But on the whole subject see Bosanquet, Logic, I, 152 foil.
4 " On the contrary it may be less so." What I meant here is this, that the residue may be less abstract than something which has been removed, or which has at least been used in the removal. But the point (however defensible) might have been omitted as superfluous.
5 " Need not always follow from the same antecedent." This state ment would, of course, be false if the sequence were pure and so " reciprocal." But here you can not assume that your premises are pure, since you are not taken to know what your " one circumstance " really is. On the Method of Difference cf. Bk. III. II. III. § 13.
6 " At all points," i.e. if induction is taken as proof.
7 There is no positive doctrine as to " Induction " set out in this work, nor had I any independent view on the subject. In the main I should have accepted, and should still accept, the view advocated by Jevons, with its two main features of Hypothesis and Verification.
JEVONS' EQUATION AL LOGIC * § i. It is pleasant, after leaving the delusions of one's youth, to find oneself in contact with something like fact. The Equational Logic has proved by its results that it has a hold on the world of reality. What works must at least be partially right. And this new theory of logic does work. One may see that its method remains inapplicable to part of its subject. One may question its convenience in certain cases, and even doubt its formula in all. But one must believe so much as this. At the lowest estimate the new system will prove what ever the syllogism is able to prove. In some points it certainly is a far more rigid test of true reasoning. It deals very easily with many of the problems which accommodate themselves to numerical reasoning. And it maintains, on the ground both of reason and experience, that, in comparison with the syllogism, it is both easier to learn and harder to forget.
In writing this chapter on equational logic, as it appears in the theory of Professor Jevons, I wish I could do two things I can not do. I wish I could give an account of the doctrine intelligible to those who have no acquaintance with it. And I wish I could form something like an estimate of its educational value and practical powers. But both want of space and want of experience compel me to a narrower and less grateful task. The object of this chapter is to ask if that account of the reasoning process which has been offered us is strictly accurate, whether as a theory it is free from mistakes. An answer in the negative will be given to this question.
§ 2. We may divide the enquiry into three main parts. In the first (A) we shall ask if propositions are identities: in the second (B) if direct reasoning consists in substitution. In the third (C) we shall discuss the Indirect Method, and with it the claims of the Logical Machine. It may prove convenient to state beforehand the main results which we expect to reach. We shall show in the first place (A) that, though every proposition does and must assert identity, yet that is not the object of all propositions. Our second conclusion (B) will be that substitution is not the real essence of reasoning, and that certain inferences will not by fair means come under this head. We shall show again that, although most arguments can be exhibited in the form of equations, yet the formula of inference which our author has given is not correct. In the third place (C) we shall argue that the Indirect Method, though perfectly valid, does not proceed by substitution: and finally we shall give our reasons for contesting a part of the claims put forth by the Machine. The reader is supposed to have made some acquaintance with the early part of The Principles of Science. § 3. (A) In asking if propositions are equations, we must remember that the sign = does not mean equal (cf. p. 23).
It denotes sameness or identity. So that the word " equa tion," which we have chosen to start with, may at once be dismissed. The question is, Do judgments consist in the assertion of identity? This point has already come before us, and great part of what follows is repetition.
1. If we dismiss all theories and look simply at the facts, then to ask that question is to answer it in the negative. How can it be said that in " Caesar is sick," or " This pond is frozen," or " Mammals are warm-blooded," we really mean to assert self-sameness? To say that, in making such statements as these, our real object is the denial of difference — that we wish to say, Although Caesar is sick he still is Caesar — is pal pably absurd. We do not wish, premising the difference, to insist on the identity. The difference itself is the information which we wish to convey.
2. If all propositions asserted mere identity, then every proposition would have to be false. If A = B and B — BC, and we go from this to the conclusion A — C, then either B makes a difference to A or it makes no difference. In the one case the proposition becomes quite false, and in the other it disappears, since B = o. How can it be true that ABC is the same as A? Is BC nothing, then nothing is asserted. Is BC a difference, then how are they the same?
Partial identities are thus all false; but simple identities will fare no better. If " =• " is taken to stand for " is the same as," then " A = B " can not possibly be true. If there is no difference, then nothing is said; if anything is said, then sameness is denied.