§ ii. (4) We have seen that, within the syllogistic limits, equational logic will work very well; and we also have seen the nature of its process. However right it is to insist that in reasoning identity is necessary, yet exactly the same must be said of difference. And I can not think that, in laying down his principle of inference and in reducing it to a formula, Professor Jevons has avoided serious mistakes.
" So far as there exists sameness, identity or likeness, what is true of one thing will be true of the other." " In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing " (pp.
Now if the " likeness " in these formulas means absence of difference, we see at once that they are tautologous or false. For so far as mere identity exists, what is true of any one thing must for that very reason be false of another. If, in the case of A, B, and C, the judgment A — C is true of A so far as A is simply the same as B, then it either is not true of A at all, or else the differences have all disappeared, and the judg ment becomes x = x. So again, if A is related to B, it is related to that which is the same as B. But " the same as B " will be simply B, and we have not advanced one single step.
§ 12. But if the formulas have another meaning, then what shall we say that their meaning is? They certainly can not mean that mere likeness will do. A need not be like C because THE PRINCIPLES OF LOGIC BOOK II. PT. II both are like B. And it is obvious that if B and C are " equivalent," A need not stand in one relation to both. Two coins are equivalent and one is in my pocket, but neither logic nor fact makes me master of the other. It is clear that this can not be our author's meaning.
The equivalence or likeness, to be that which is meant, must exist to a sufficient extent or degree. But what is the degree which is sufficient? " The general test of equality is substitution" (Principles, p. 19). But here again our ques tion is not answered. It would never do to say, you may sub stitute when you have a sufficient degree of likeness, and that degree again consists in your ability to make a substitution. And this is not what is meant. What I think is meant is that a certain amount of likeness will give conclusions, and that, when you can substitute, you may know it is there. But I do not think that Professor Jevons has anywhere told us in what that degree itself consists.
§ 13. Still I think he has given us the materials for an answer. The question we have before us is this: Given a term B in relation with C; or otherwise, Given C as what is true of B, then what amount of sameness between A and B will warrant us in writing A for B? The first answer to be given is that no amount is wanted. There is not the very smallest need for A and B to be like or equivalent. But the second answer to be given is this: the sameness required is the sameness of the one subject. If A and B are both qualities of X, or again if B is a quality of A, then A and C will be interrelated. The quality of the subject is the middle term, whose predicates in some way qualify the subject. Or the identity of the subject 4 is the middle term and, so far as this identity extends, the attributes must all be related and con joined.
We have finished our examination of the theory of propo sitions, and also of reasoning by substitution. We come now to a third and most important point, the question of the Indi rect Method and the Logical Machine. I will anticipate briefly the result we shall reach, (a) The essence of the Indirect Method is a process which can not possibly be reduced to substitution, (b) In part of that process substitution may be used, but another form of reasoning is just as applicable, CHAP. IV JEVONS' EQUATIONAL LOGIC 379 (c) The Machine will not really give complete conclusions.
(d) It is improperly limited to one kind of reasoning.
§ 14. (C) (a) The Indirect Method is a process of ex clusion. In using it you must first find all the possibilities, and then by removal of the rest you leave only one. In other words, you have a disjunction, and remove all alternatives except a single remainder. Because the subject, if taken as real, must be taken as fully determined and particularized, therefore the remaining possibility is real (cf. Book I. Chap. IV. ). A is b, c, or d, it is not b or c, it therefore is d. This is the essence of the Indirect Method, and we already have to some extent made its acquaintance.
§ 15. We know that this process falls outside syllogism. And from that we might argue at this stage of our enquiry that it can not be reduced to substitution. But if it can not be reduced to substitution, Professor Jevons' best work contra dicts his theory. Let us see how he tries to avoid this conse quence.
" The general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred. Now this result clearly follows from our process of substi tution; for if we have the proposition — and we insert this expression for A on one side of the self- evident identity Ab = Kb, we obtain Ab = AB& -|- AbC -|- A&D; and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction; there remains Ab=AbC-\-AbD.
Thus our system fully includes and explains that mood of the Disjunctive Syllogism technically called the modus tollendo ponens" (Principles, p. 77).* But this, I think, will not stand a moment's examination.
*I may remind the reader that •[• here means "or," and b means " Not-B." I do not use these signs in the text.
380 THE PRINCIPLES OF LOGIC BOOK II. Pi. II In the first place the operation of striking out one part and asserting the rest is the essence of the method, and yet it is not even in appearance reduced to substitution. In the second place in this example the reasoning by substitution is perfectly useless. It does not bring you one step on your way towards the conclusion.
I will take a perfectly simple instance. " A is b or c" and " A is not b" These are the premises, and from these I should say that you go directly to the conclusion " A is c" Professor Jevons, if I understand him rightly, contends that you go through a process of substitution. A = b or c, A = not-b. Insert the expression for A, "A is b or c" on one side of A not-b = A not-b. Then A not-b, = A not-b and b or A not-b and c. But A not-b and b = o, therefore A not-b = o or not-b and c.
But surely, if words have any meaning, when I know that A is b or c, and that A is not bf I do know at once that b must be removed. And, on my removing b by an ideal experiment, c by itself is what I have left. If I please I may write this " c or o." But I really can not perceive what advantage I get by turning in a circle to come back to my starting-place. A is b or c, and it is not b. If possible how ever let A be b. But, if it is b, it will be b and not-&. That is impossible, and therefore follows — what? Why simply that A is not b. I have used the premise to prove itself. And, if in answer I am told that this is not so, for I have enriched what was given me by the alternative " or o," then it seems to me that I may fairly reply, If you do not know, given only b and c, that when b is gone, c is what is left behind; then how on earth can you tell that, given " c or o," when o is gone, c is all that is left? I confess to me one is no clearer than the other.
§ 1 6. What I think has occasioned this complete mistake is an erroneous idea as to indirect reasoning. For that we must have a disjunction to start with, and by removing one member we prove the other. And we generally have to use direct reasoning downwards. We assume as one of our premises that alternative which we want in the end to get rid of, and on this assumption we bring out a conclusion which contradicts something contained in the premises. This is the CHAP. IV JEVONS' EQUATIONAL LOGIC 381 usual course, but it is not more than usual. Direct reasoning downwards is not always wanted. For when the premises themselves give the removal of one alternative, what more can we prove by such direct reasoning? We have in our hands not only the disjunction, but also the exclusion of one alter native. Where direct reasoning is required it is simply pre liminary to the final operation, and is wanted merely to prepare the subject; and when the premises give the subject ready prepared, what is there which we possibly can have to wait for?
And I think this mistake is connected with another. I suspect that an error as to the Laws of Contradiction and Excluded Middle has helped to lead our author into this pitfall. But when we know that the Law of Excluded Middle 5 is one case of disjunction, and in no sense the basis of it (Book I. p. 151), we see at once that no mystical force arises from the proof of a self-contradiction. If we get to that by turning in a circle, the end will hardly justify the means. It has no power to absolve our consciences from the ordinary sin of logical fallacy.^ I must not be considered as wanting in respect, if I illustrate what I mean by another instance. Suppose that my premise is " A is b." Will any one deny that to prove from this that " A is b " is a frivolous circle? But it is easily done. For, if possible, suppose that A is not b; then A will be both b and not-&: or insert, on one side of the self-evident identity A not-fc = A not-&, the expression for A. Then A not-& = A not-& and b. As one side of our equation is now self-contra dictory, we strike it out according to the law of contradiction, and then there remains A not-& = o, or A is b. I must be allowed to state my conviction that this circle is the same as what we had above. In both cases alike the premise has been used to bring out nothing whatever but that which it gave.
The Indirect Method, we so far have seen, can not be reduced to a process of substitution.
§ 17. (b) If we consider that Method as employed by Professor Jevons, it does make use of the equational form, but there is no real necessity for its so doing. This process consists of the following four steps.
"i. By the Law of Duality develope the utmost number of 382 THE PRINCIPLES OF LOGIC BOOK II. Px. II alternatives which may exist in the description of the required class or term as regards the terms involved in the premises.
2. For each term in these alternatives substitute its de scription as given in the premises.
3. Strike out every alternative which is then found to break the Law of Contradiction.
4. The remaining terms may be equated to the term in question as the desired description " (Principles, pp. 89-90).
The one part of this process which employs substitution, we see, is the second. But it is performed just as well by the ordinary method. All the possible combinations of the terms are given us, and our object is merely by means of the premises to remove those combinations which the premises contradict. In what shape then ought we to have our premises? Surely one would say in the shape of -combinations. It is just such combinations that the ordinary process would give us directly, and we get them by substitution in a roundabout way. For the " description " of the term is, as we saw, the judgment we make about the term. Hence this part of the method, as employed by Professor Jevons, is valid just so far as it can be stated syllogistically. For the premises are combinations of attributes. They are related, as Professor Jevons says, " just as the qualities of the same object " (ibid. p. 114); and if they were anything else, his method could not deal with them. We can combine them directly, if we please: and it is simply our choice, and perhaps sometimes our convenience, if we combine them from behind through their common subject.
Thus we may use substitution to prepare for our conclu sion. But we can not use it to draw that conclusion. Its operation ends with the second step.
§ 18. We see, from examining the method itself, that it deals with syntheses or combinations, and does not deal at all with equations. And the method, as practically worked with the machine, confirms the truth of the view which we have taken. Professor Jevons himself with the greatest can dour has called attention to this consideration.
" It is no doubt a remarkable fact that a simple identity can not be impressed upon the machine except in the form of two partial identities, and this may be thought by some logi- CHAP. IV JEVONS' EQUATIONAL LOGIC 383 cians to militate against the equational mode of representing propositions" (Principles, 112).
It would be to me even more than remarkable if the ma chine could work with simple identities. But the fact, which Professor Jevons rightly finds remarkable, has I think a still more remarkable counterpart. The conclusions of the machine, if I understand them properly, contradict one another when read as equations in the sense of assertions of simple identity. A — B — C is consistent with Not-A — B — C; 6 but how can we reconcile A = C with C = Not-A?
§ 19. (c) We come now to the subject of the Logical Machine, and we have to enquire what work it performs. Of the mechanism employed I have no knowledge. I am so incompetent to say anything about it, that I can not have the pleasure of congratulating Professor Jevons on what I must believe is no small achievement. But what the machine does perform is this. All the possible combinations of the terms are worked out, and are lying ready drawn up in the machine. The operator puts in at one end his premises, each in the shape of a combination. The combinations of these premises remove, each one, all the possibilitites with which it is irrecon- cileable. And what comes out, so to speak, at the other end of the machine is all the residue of possible combinations which have not been so excluded by the premises. It is easy to exaggerate the powers of the machine. But I think it is impossible to deny that it executes such work, as must other wise be done by a process of thinking. For myself I do not hesitate to say that it performs mechanically an operation which, if performed ideally, would be an inference. And in this sense I think Professor Jevons is justified in his claim to have made a reasoning machine.7 Apart from the practical utility of the instrument, which in certain cases may be con siderable, we must admit that, from a merely theoretical point of view, it is a most interesting and instructive phenomenon. If Professor Jevons had made no other contributions to logic, we might yet be sure that his name would go down with the history of the science.
But to say on the other hand that the machine will execute the whole process our minds perform in the inference— that the raw material goes in at one side, and the finished con clusion comes out at the other, would be travelling far beyond 384 THE PRINCIPLES OF LOGIC BOOK II. Pi. II the fact. Before the premises can be worked on the instru ment, they have of course to be reduced and formulated, so as to take the shape of combinations of letters. But this is not the most important point. The result that comes out and is presented by the machine, is not really the conclusion. The process is not finished when the machinery stops; and the rest is left to be done by the mind. What is called " reading " the conclusion is to some extent making it.
§ 20. I will explain what I mean. In the machine is drawn up a complete disjunction of the possible arrangements of those terms which we employ. Before we begin to work the problem the machine thus supplies us with one of our premises. It states all possibilities, and this is its strength. But it states mere possibilities, and this is its weakness. We begin our operation, and insert the combinations which are given us by our data. These combinations are the rest of the premises. The machine, as it receives each combination, re moves from the list of all the possibilities those which are inconsistent with this datum. Then the remainder of the possible combinations are exposed. But they still remain bare possibilities, and are never stated as actual facts.
The process may be taken as having five parts, i. The complete disjunctive statement of possible combinations. This is given ready-made by the machine. 2. The reduction of the premises to the shape of combinations. This is done entirely by the operator. 3. The discovery of those alterna tives which are inconsistent with the combinations of the premises. This step is performed entirely by the machine. 4. The removal of those alternatives. This step again is performed by the machine, and it is the first part of that final inference which gives the conclusion. 5. The assertion that what is left is true, and that, if but one possibility remains, that is fact. This is absolutely necessary to complete the inference, and this is done entirely by the operator.
The final step may seem to some persons a final super fluity. But on that view of the nature of reasoning by way of the exclusion of alternatives which has seemed to me true, it is integral and essential. Yet it can not be said to be performed by the instrument.
§ 21. I wish to stand on this statement of the case. But CHAP. IV JEVONS' EQUATIONAL LOGIC 385 it is possible to use also an argumcntum ad hominem. If the too undiscriminating friends of the machine assert that its result is a categorical statement, they can hardly fail to com promise it deeply. They will make it an instrument for the production of falsehoods. Let us take one result that is given by the machine (Principles, 109).
Now, there being here but one possibility, if A is assumed, we are practically safe in contending that the machine cate gorically asserts this one possibility. But, suppose we take the same line throughout, we plunge at once into a sea of non sense. Contradictory possibilities can co-exist as long as they remain mere possibilities, but the moment you affirm them as actual fact, they exclude one another. And, if so, either the machine brings out false conclusions, or all must be read as mere possibilities. You have no warrant from the machine for the assertion A is C. A may be C; and because it may be, and because there is nothing else that it may be, and because you know that it must be something to C one way or the other, you therefore infer that A is C, a conclusion not given to you by the machine.
§22. (d) The machine performs more than we have a right to ask, and it is a pity to credit it with fictitious powers. We have seen that it does not bring out a conclusion. But it is limited beside in another respect. Although it does not work by substitution, yet its range is limited to that kind of inference which is possible in equational logic or in syllogism. It can not deal with any other combinations than those which repre sent the co-existence of qualities within a subject. And this is a very serious defect; for it means that the machine refuses to touch more than a part of the subject.
This is not the fault of the Indirect Method itself. Apart from restrictions artificially imposed on it, that is applicable everywhere and to all kinds of matter. If my premises are " A is to the right of B, and B of C," I may go directly from these to my conclusion; but, if I choose, I may use the i direct method. The possibilities of A with respect to C are 386 THE PRINCIPLES OF LOGIC BOOK II. PT. II either absence of any spatial relation, or A to the right of C, or to the left of C, or neither and above it, or below it, &c. But the premise " A to the right of B," will exclude (as we should see by an ideal construction) every alternative we can find other than A to the right of C. For, if we assumed any one of the others, we should bring out a result incompatible with our premise. The remaining possibility is therefore fact. This is perfectly familiar and common-place reasoning, and a system, in which it can find no place, must assuredly be called at least incomplete.
§ 23. The result of our perhaps too brief examination may be stated as follows: — 1. The Indirect Method has absolutely no vital connection with the Substitution of Similars.
2. That Method itself is flawless and complete, but as used by Professor Jevons it is improperly limited.
3. The machine which works within these limits will not actually give a categorical conclusion.
4. These unfortunate limits are also those of equational reasoning.
5. They coincide exactly with the boundary of the syllo gism, and a large part of reasoning falls entirely without them.
6. The method of Substitution is syllogism upside down, and its principle has not been accurately formulated by Pro fessor Jevons.
I must leave this subject with an expression of regret. I am sorry to have had no more space available; and I am sorry to have dwelt almost wholly on those points in which I am unable to follow the author. It would have been more pleasant, if it had been possible, to have called attention to the various merits of his logical work. But still, even if my praises could do him any service, fortunately he does not stand in need of them. I may end this chapter by expressing my belief, that no living Englishman8 has done one half the service to logic that Professor Jevons has done. No living writer, to the best of my knowledge, now Professor Lotze is dead, has done more. Personally to myself, and so far as my own studies are concerned, Professor Jevons' book has been of very great use; and I could not truly say that of any other English Logic. It is not inability to accept conclusions which CHAP. IV JEVONS' EQUATIONAL LOGIC 387 prevents one learning. And there can not be any one who has left unread the Principles of Science, who has not something to learn from it.* * Since this chapter was written Professor Jevons' lamented death has taken place, and has deprived me of any opportunity I might other wise have had of learning from him in what points I have failed to understand his doctrines. I have thought it best to leave the chapter as it stood.
But there is another point on which the reader may look for some explanation. He may ask why I have failed to examine one of those views of Equational Logic which treat the subject mathematically. And I am compelled to throw the burden of the answer on those who had charge of my education, and who failed to give me the requisite instruction. It would have been otherwise a pleasure to have seen how the defects of the Equational theory appeared in a mathematical form. For, at the risk of seeming no less prejudiced than ignorant, I am forced to state the matter so. If I knew perhaps what Mathe matics were, I should see how there is nothing special or limited about them, and how they are the soul of logic in general and (for all I know) of metaphysics too. Meanwhile I may suggest to the mathe matical logician that, so long as he fails to treat (for example) such simple arguments as " A before B, and B with C, therefore A before C," he has no strict right to demand a hearing. Logic is not logic at all if its theory is based on a previous mutilation of the facts of the subject. It may do something which perhaps is very much better, but it does not give any account (adequate or inadequate) of reasoning in general. And at the risk of exhibiting prejudice once more, I may say that this consideration seems to me to be vital.3 ADDITIONAL NOTES 1 On the subject of this Chapter see the Notes on Book I, Chap. VI, and also T. E. III.
2 "Single subject," "self-same subject." Cf. Bk. I. VI. § n and T. E. III.
3 " Though the statements are false." These words would, I think, have been better omitted. The " four meanings " are as follows, (i) Every statement contains a diversity, but (ii) its diversity does not make it two, so that by dividing it you can get rid of the connected unity which makes its character — here of wilful falsity. And the essence of its character, while (iv) remaining throughout one and the same, is yet (iii) affected by, and made more intense by the number of its instances.
4 " The identity of the subject." See Note 2.
THE PRINCIPLES OF LOGIC BOOK II. PT. II s « Excluded Middle." See Bk. I. V, Note 12.
6"A — B — C is consistent, etc." It would be better before "is" to insert "(as commonly understood)"; for, if A is taken as pure, i. e. as unconditional, the above statement would be incorrect. Cf. § 21, and see the Note on Bk. II. II. III. § 12.
7 " A reasoning machine." Dr. Bosanquet (K & R, pp. 327 foil., and Logic, II, 150) has called attention to the point that all instruments of measurement and observation have a right to be called " reasoning machines."
8 " No living Englishman." This was of course published in 1883, and I think that it was true. My eulogy may perhaps on the whole be exaggerated, and that question I leave to others to decide. What I wrote remains as the expression of the gratitude I felt towards one whose book had helped me greatly in my logical struggles.
9 The second paragraph of this foot-note would have been better omitted. When writing it I did not know of the existence of a mathematical logic which was not equational. But even now I am in effect perhaps in no better case.
Whether a student of logic, who is incapable of learning mathe matics and has therefore to leave out of his theory a recognized part of the facts, should never have written on logic at all, or should later at least suppress all that he once wrote — I will not offer to discuss. And what should be his attitude towards a claim to base the principles of logic on mathematics, I once more hardly know. If a person like myself ventures to point out that something of what is thus offered seems to himself to be untenable and irrational — he can be met with the reply that, if he understood mathematics, he would forthwith think otherwise. And what his answer to this should be, I confess I can not say.