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.
366 THE PRINCIPLES OF LOGIC BOOK II. Pi. II 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 CHAP. Ill THE INDUCTIVE METHODS OF PROOF 367 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.
368 THE PRINCIPLES OF LOGIC BOOK II. PT. II 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.
ADDITIONAL NOTES 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 CHAP. Ill THE INDUCTIVE METHODS OF PROOF 369 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.
CHAPTER IV 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 propo- CHAP. IV JEVONS' EQUATIONAL LOGIC 371 sition 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 372 THE PRINCIPLES OF LOGIC BOOK II. Pi. II no difference, then nothing is said; if anything is said, then sameness is denied.
3. It is obvious, if we are to keep to identity, that sub ject and predicate must be wholly the same. AB = AB, ABC = ABC. But even here it is doubtful if we can stay. For even when we reach a tautologous statement we have still a difference in the position of the terms (cf. Book I. Chap. V.). If we wish to be consistent even that must go. We must take one side of our former reduplication; we must say, for instance, AB or ABC. In that, having given up our search for identity, we suddenly find the whole content of our asser tion. Assume AB, then A is B. Assume ABC, then A is C. In our seeking to get an equational truth, we got all the dif ferences together on each side. But the synthesis of these differents was just what we really wanted to assert. Strike out one side, and strike out the " =," and we have the content of the whole judgment.* Assertion is not confined to the affirmation of sameness, and identity and equality are but one kind of predicate. If we use the language of the traditional logic, then in " S = P " the " = " has nothing to do with the copula: it falls entirely within the predicate, and " A = AB " is " A— = AB." If we wish to say that A is equal to or the same as B, the natural mode is, I think, to say that A and B are the same or equal. If we will not do that, and so openly admit the existence of difference, we must come in the end to " A = B," on the left hand side, is just the same as " A = B," on the right hand side. And since the sides are different even that is not true. § 4. The foregoing section merely asserts that a difference is affirmed by every proposition. Judgment can not be reduced to one-sided identification. In the attempt to reduce it we found that we got the whole matter of the judgment on each side of the copula. Thus in " sodium = sodium metal con ducting electricity " the judgment falls on the right hand side. The assertion consists in the synthesis with sodium of the being a metal and conducting electricity; and, when we know that, the " sodium " and the " =," of the subject and copula, are false or meaningless. You say that it makes no difference * We are not dealing here with " simple identities." For them CHAP. IV JEVONS' EQUATIONAL LOGIC 373 to sodium that it is a metal and conducts electricity. That surely is a rather odd method of saying that there is no difference whatever to make, and a still more eccentric method of implying that this makes all the difference to sodium.
§ 5. No proposition asserts mere identity, but without the statement or implication of identity no judgment can be made. The solution of this puzzle, which the end of the foregoing section hints at, is that sameness and difference imply one another, and are different sides of the self-same fact. Mere identity or difference is therefore unmeaning. And hence, although it is false that in judging we always mean to identify the subject and predicate, yet in every judgment an identity can be found. For where sameness is asserted difference is presupposed. Where difference is asserted there is a basis of sameness which underlies it. And it follows as a consequence that, if you do not mind your implications being put on a level with your meanings, you can show every judgment in the form of difference united by identity.
§ 6. For in every judgment the differences joined may be taken as the qualities of a single subject2 (cf. p. 27, and p. 180). In " sodium = sodium metal" we assert that within the subject called sodium the attributes sodium and metal are conjoined; and if you please you may express this by saying, that, under the differences sodium and metal, there is yet no change from one subject to another. Again, in " Equi lateral triangle = equiangular triangle " what I mean to say is that, despite these differences, you still have one and the same triangle, or again that, if one of these qualities exists, you will have the other in the self -same subject. Take again " The Pole Star = the slowest-moving star: " this means either that one star possesses these two differences, or that, in spite of these differences, the star is the same. In every case we have identity and diversity, and, though we accentuate one or the other, yet in every case both must co-exist.
I will illustrate the foregoing by other instances. Take " These fifteen statements are every one perjuries." The identical subject is here either each statement or the quality of perjury which appears in each. There are hence four meanings. In the first I assert that in every statement perjury must be added to its other qualities. In the second 374 THE PRINCIPLES OF LOGIC BOOK II. PT. II I deny that, though the statements are false,3 we have any right to abolish the perjury by making thirty statements out of fifteen. In the third I complain that a single crime has occurred with fifteen different sets of details. In the fourth I refuse to admit the diversity of the fifteen qualifications as any proof that the crime is not the same.
Or take the instance of equality or sameness itself. When I say that A and B are equal, I assert that in the differents A and B their quantity x is for all that the same. If I say " A and B are precisely the same," I must first take A and B as differenced by place or time or some other particular, and then against that assert their identity. The equality in one case and the sameness in the other may be treated as the subject in which A and B co-exist as attributes.
If the doctrine already put forward is true, there can be no such things as " simple identities." " Equiangular triangle = equilateral triangle " is false if it denies the difference of quality, or is false if it ignores the distinction of subjects. The identity it asserts must exist under differences. Thus among triangles the subject of equilateral is one and the same with the subject of equiangular. The natural way to state the fact is to say, The different subjects are the same, or The diverse qualities imply one another.
§ 7. The result of our enquiry as to propositions is not of good augury for the doctrine of Substitution. True we find that all subjects assert an identity, but then they no less assert a difference. Our sign " = " has turned out quite inapplicable. If S and P are made quite identical, the judgment disappears or falls only on one side. If again S and P are allowed to be different, the sign of identity asserts a falsehood. This so far is ominous. It is ominous again that every identity can be shown as the connection of attributes within a subject. And there is another omen we have not yet noticed. All judg ments, we long ago have found, can be understood as assertions of identity. But the class of relations in time and space, it appears, are not amenable to the Method of Substitution, or at least in public decline to appear so (cf. Book I. p. 22). I can not but think that with such auspices against it any cause must be lost.
§8. (B) We come now to the second branch of our sub- CHAP. IV JEVONS' EQUATION AL LOGIC 375 ject. Does the process of reasoning consist in substitution? The foregoing has shown that this is not possible.
(i) The terms which we substitute must be the same: but if the same then you can not substitute. If your process does not give you a difference, it is no process. If it gives you a difference you have broken the identity. Thus if reasoning consists in substitution, its essence lies in the substitution of differ ents.
Let us take as an example, " A is equal to B, and B to C, and therefore A to C." It is impossible here by substitution of identicals to come to any conclusion whatever. For what is there identical? A is not the same as B, nor B as C, nor is " equal to B " the same as A. The identity really lies in the quantity of A, B, and C. The quantity of A and B is the same, and so is that of C and B. The quantity therefore of A and C is the same. But you can not show this by substi tution. For in the quantity of each there is no difference. The terms are x A, x B, x C. Now if you substitute x A for x B, you substitute things which are not the same. But if you substitute mere x, you do nothing at all, for already you have the term x B. A is equal to B, but it is not the same. The quantity is the same, but it is one and not two.
The real process of the reasoning consists in connecting the differences A and C on the basis of their common identity x. It may also be stated as a substitution. Take x with any one of the differences, and substitute x with any other differ ence. The differences then found co-existing in x will be the conclusion which we require. But this substitution is a re placement by differ -ents.
§ 9. (2) Substitution, so far as it works at all, is an indirect method of synthesizing differences. The rule is to substitute the " expression " for the term. But the " expres sion " is the judgment about the term. The rule then says " Substitute the judgment for the term." In other words, a term will not do; you must have a premise, and that means a judgment. You must leave your identity and get to dif ferences.
In "sodium is metal and conducts electricity" (§4), sodium-metal takes the place of sodium, and metal gives way to metal-conductor, and we say this makes no difference to 376 THE PRINCIPLES OF LOGIC BOOK II. PT. II sodium, or sodium is the same with all this difference. But the real subject, which remains the same, is something which underlies these differences; and the real process is the addi tion of difference which developes the connection of attributes in this subject. It is entirely to mistake our object in view if, while we try to get the synthesis of diverse attributes, we talk as if all we wanted was to keep the identity of the subject. It is simply to stand the process on its head, if we make every step by uniting differences, and then speak as if throughout we had done nothing but remove them.
" Substitute for the terms their expressions," that is in other words combine the premises. It is an artificial way of performing the old task. For reasons which I can not here enter into, the artifice in some cases is very useful. But it is simply the syllogism turned upside down, and it is confined to the same insufficient limits.
§ 10. (3) The method of Substitution has set itself free from some of the superstitions of the traditional logic. For certain purposes it is far more useful. Everything again that can be proved by syllogism can also be proved by its modern rival. But on the other hand Substitution will prove nothing that can not be shown by syllogism. The limit of both is precisely the same. They are confined to the relation of sub ject and attribute and the connection of attributes within a subject; and beyond that category neither will work (cf. Bk. II. Parti. Chap. II. §6).
To prove syllogistically that, because A and C are both equal to B, they are equal to one another, is quite impos sible.* But it is just as impossible to prove the conclusion by substitution. The premises you have got are A = A equal to B, B = B equal to C; and the quaternio terminorum can only be avoided by taking the premises in a sense which is false.
It is needless to repeat against the equational logic the * " Quantity of A is the same as quantity of B, quantity of B is the same as quantity of C, and therefore quantity of A is quantity of C " will not do at all. If the quantity is taken in abstraction then it certainly is the same, but you can not show from that that A, B, and C are related as equals or related in any way. But if you take the quantity in its relations to A and B and C, in that case you have quaternio terminorum, or otherwise the premises become false. The relation of equality never could be got out in the conclusion.
CHAP. IV JEVONS' EQUATIONAL LOGIC 377 objections we have urged against the syllogism. If a logic will not deal with the syntheses of degree, of space, and of time; if even, as we shall see, its own Indirect Method falls outside its boundaries, then that logic does not give the true method of reasoning. It is not made too narrow because it requires an identity underlying the terms of its premises. It is made too narrow because in its conclusions it is confined to the category of subject and attribute. In a remarkable pas sage (Principles, Ed. II. p. 22) I understand Professor Jevons to admit these limitations. His logic, so far as it exists at present, appears to be confined to " simple relations." " A simple logical relation is that which exists between properties and circumstances of the same object or class." But, if that is so, then the theory of reasoning will cover only one portion of the facts.