two denials there is no conclusion. If one premise denies and keeps to denial, then one premise at least is limited to the genus of subject and attribute. If the middle term B falls out of the conclusion, if A and C are connected through B, but not by means of an intermediate B, then the conclusion denies and falls also within the above-named genus. But if B is kept standing, the conclusion may at least in part be positive, and is not confined to a single category.
The general formula for negative reasoning, if we confine ourselves to the side of bare denial, may be stated as follows: 5 If B repels a content C, and is in relation with a third term A, then A and C will either be related directly by way of denial or else will be elements in a whole A — B — C, of which at least one member will be confined to the genus of subject and attribute. And I think with this we may take leave of a subject which has proved perhaps more troublesome than in teresting.
ADDITIONAL NOTES ADDITIONAL NOTES 1 The statement that all reasoning, negative as well as positive, depends on an ideal whole, and that this whole can be called a con struction, is so far correct. But otherwise this section, and much of what follows, is unsatisfactory. Every negation (see on Bk. I. Chap. III.) implies a disjunction. And only because, and so far as, negative reasoning is based on and further developes a disjunctive totality and system— does it possess a real value. For an admirable exposition of this view the reader is referred to Bosanquet's Logic.
If we keep to mere denial, what is denied will certainly fall some where else in the Universe, since no mere ideas are possible. But, because the variety of special worlds within the Universe is indefinite, and because the merely denied is not, so far, located, you can base no special connection on the fact of mere simple denial. If negation is to be fruitful, it must (to repeat this) stand upon and move within a scheme of specialized alternatives, related to each other at once as positive and negative.
Hence it is scarcely worth while for me to attempt to correct chapter in detail. I will, however, touch on a certain number < points.
2 The usual demand for the elision of the middle term seems n defensible, and any rule that the conclusion must merely deny shou therefore be modified. See §8. But the rule which condemns 284 THE PRINCIPLES OF LOGIC BOOK II. PT. I two negative premisses, in the sense of two denials, must stand. For what is denied may fall in worlds not so connected as to make a construction possible. Hence, unless by going beyond mere denial one premiss becomes positive, no conclusion can be reached. In §4 after quaternio terminorum " we should add " or else one positive premiss."
3llf you keep to mere denial, as distinct from exclusion, repulsion or absence, all that is implied is an unspecified whole (x) containing two diversities (A and B). These must be positive, but, so far as you merely deny one of the other, you attend simply to their differ ence. Further, by identifying one of them (A) with C, you can deny the other (B) of C. But neither here nor elsewhere is there any inference through mere denial beyond the category of subject and attribute. As soon as you have assumed worlds containing arrangements and relations other than those of identity and difference, you have gone beyond mere negation in the sense of denial.
Hence the "general formula" (§9) can not stand, and should perhaps be read thus — " If you deny of B a content C, C can also be denied of that which is identical with B, and can further be related indirectly by denial with that which is related positively to B." But, though in the latter case the " conclusion " need not be " con fined to a single category," the inference, and what actually is concluded, never goes beyond the category of subject and attribute. Statements to the contrary (§§ 2, 8, and 9) are erroneous.
4 In the way of minor corrections I may here note that we should insert "definite" before "existence or position"; and (at the end of the paragraph) should read "move in any one real world at all." And, generally, I would remind the reader that such terms as "re moval," " exclusion," " repulsion," and even " absence," all are affirma tive in the sense of at least containing a positive aspect. And this aspect goes beyond what is contained in negation, if and so far as we take that as mere denial.
6 For " the general formula " see Note 3.
CHAPTER VI TWO CONDITIONS OF INFERENCE § I. We may briefly recapitulate the result we have reached. An inference is always an ideal construction result ing in the perception of a new connection. So far as this perception of the conclusion is concerned, there is no possibility of laying down rules, and the syllogistic logic teaches a super stition. That logic again has failed to include all the prin ciples of synthesis which operate in construction, and it is falsely confined to a single category. It is wrong again as to the number of the premises; and, in insisting on the neces sity of a major premise, it is clinging blindly to exploded meta physics in direct defiance of the most palpable facts. And it makes a further mistake as to the necessity of elision.
It might seem that having thus rejected the syllogism we must throw in our lot with its hereditary enemies. But yet, if the friends of the syllogism will allow it, we would rather take a place on their side. Our differences are trivial com pared with our agreements, and as against the enemy our cause is the same, for we have in common these two beliefs: (i) It is impossible to reason except upon the basis of identity, (ii) It is impossible to reason unless at least one premise is universal. It will be time to say vlcerunt empirici when these positions have both been forced.
§2. (i) I will begin with the necessity of an identical point. We know that an inference is an ideal construction, and the reality of this construction depends on its unity; if the construction is not individual it is merely fictitious. But how can any construction have unity unless it is united by a common point? And how can any point be common, unless in both the premises it is one and the same?
It is obvious that suppose the problem before us is to find the relation of S to P by means of their common relation to M, and if, by the hypothesis, S-M and M-P must be given separately, an advance is impossible, unless in both premises 286 THE PRINCIPLES OF LOGIC BOOK II. Px. I M is the same. Given S — M1 & M2 — P you can make no construction, for you have no bridge to carry you over from M1 to M2. The back of your inference now is broken and the extremities no longer belong to any individual principle. Un less M in both cases is absolutely the same you can not inter relate S and P.
If we are willing to give up the superstition of the copula and to admit a diversity of relations in judgment, we may say that in inference every pair of premises has one term the same, and that, if it is not the same, there can be no inference.
§ 3. It is obvious, if we dismiss our hardened prejudices and consider the question fairly by itself, that you can not argue on the strength of mere likeness.1 Whatever else may be right this at all events must be wrong; " A is similar to B, and B to C, and therefore A is like C," is a vicious infer ence, one that need not always be mistaken in fact, but that always must be a logical error. In practice I think we should all admit this. An inference based on nothing but likeness is utterly invalid; it is certainly ambiguous and probably false.
Likeness and sameness should never be confused, for the former refers properly to a general impression. Similarity is a perceived relation between two terms which implies and rests upon a partial identity. If we say that A and B are alike, we must be taken to assert that they have something the same. But we do not specify this point of sameness, and the moment we do that we have gone beyond mere similarity. If A and B for instance both have lungs or gills they are so far the same, and, on the strength of and because of this partial identity, they may present themselves to us as generally similar. But now add to these the further statement " B and C are alike." If we reduce the likeness here to partial identity we may find that the common point is here once again the possession of lungs or gills, and on the strength of this we may go on to argue that A and C (the extremes) are alike. But what actually interrelates A and C is not general similarity at all. If all you knew was that B was like C, the point of identity would be quite unspecified, and the fact might be, not that both had lungs or gills, but that each had one eye or the freedom of the will. In this case though each CHAP. VI TWO CONDITIONS OF INFERENCE 287 pair has its own internal likeness, you could not infer the similarity of A to C.
And if in answer I am told that this is irrelevant, and that it does not apply where the likeness is exact, I can only reply that I am waiting, and have been waiting for years, to be told what is meant by an " exact likeness." " A and B are not the same, but they are exactly alike, and therefore whatever is true of B must be true of A." But what can this mean? In the case of some twins it might be right to punish one for the other, and we should no longer care to identify criminals. If a picture is " exactly like " a person, then if one is not dead the other will be alive. If a cast is " exactly like " an original I suppose the same thing will be in two places at once; and it is no mere metaphor if in certain cases the father is said to survive in his children, though the children might then cease to survive the father. But it is idle to pursue these frivolous consequences; the meaning which "exactly like" carries to my mind is nothing whatever but "partially the same" or "identical in some point or points." Likeness is always a perceived relation based upon a partial identity. In mere general similarity the identity will be indefinite; where the likeness is more special it must at least be partly defined, and where the similarity is called "exact" I understand that there is a definite point or points, in respect of which the same ness is complete. And if likeness did not imply identity all inference based upon it would be vicious. In practice every one would allow it to be vicious, nor do I understand how in theory it is possible to take it as having any other character. I am most anxious to enter into (if I can), and to discuss the meaning our "advanced thinkers" may have attached " likeness " or " similarity." But I am forced to say again in this place what I had to say elsewhere some years aga While our "advanced thinkers" merely sing the old song which they have learnt and which their fathers have taught them, they can hardly expect to have its meaning discussed nor can they complain if they are treated as having no construction of given premises is not possible un- less each pair of premises has a common point. And * Ethical Studies, p. 151 (Ed- IL P- 288 THE PRINCIPLES OF LOGIC BOOK II.?T. I common point must be an identical term. Thus in "A — B B — C therefore A — B — C" the B in each premise must not be merely alike, but must be absolutely the same. But here, after having avoided one error, we are threatened by another and opposite mistake. For if it is wrong to say that B is not the same, it is equally wrong to deny that it is different.
This may look mysterious but is really quite simple. If B in both premises were so far the same that no difference of any kind belonged to it, then it is obvious at once that both premises must be identical, or else that their differences do not concern B. But in each of these cases the inference dis appears. If the premises are the same their repetition is meaningless, and if the differences they contain are indifferent to B it is clear that no construction can be made, since, if B is the centre, it carries no radii and has no circumference. An identity which is not a synthesis of differences is plainly inert and utterly useless.
B is the same amid difference, and though different is the same, for it is an ideal content, the product of abstraction, appearing in and differenced by two several contexts. So far as it is the one content B, so far it is absolutely and entirely the same; so far as it is a member of diverse connections, so far it carries with it a difference. And the process of inference depends entirely on this double aspect; for it is because B is different and yet the same, that its differences are able to be interrelated. If it were not different it would have nothing to connect, and if it were not the same there could be no connection. Inference rests upon the assumption that, if the ideal content is the same, then its differences will be the radii of one centre. In other words if B is the same, what is true of it in one context is true of it in another.
§ 5. We have returned to what we called the Principle of Identity (Book I. Chap. V.). We might call it again the Axiom of the Identity of Indiscernibles, and we can put the thing in more simple language if we say that inference rests on the principle that what seems the same is the same,2 and can not be made different by any diversity, and that so long as an ideal content is identical no change of context can destroy its unity. The assumption in this principle may be decried as monstrous, and I do not deny that perhaps it is false. In a CHAP. VI TWO CONDITIONS OF INFERENCE 2§Q metaphysical work this question would press on us, but in logic we are not obliged to discuss it (Book III. Part II. Chap. IV.). The axiom may be monstrous or again it may be true, but at least one thing is beyond all doubt, that it is the indispensable basis of reasoning. It may be false meta physically, but there is no single inference you possibly can make but assumes its validity at every step.
§ 6. It is easy to misunderstand it, and it is sure to be misunderstood. I shall be told that spaces and times are indiscernible and yet are not identical. But this objection rests on a complete mistake. As spaces or times of a certain character A and B surely are identical; as different elements within the same series A and B are surely not indiscernible. It is one superstition to think you have relations whose terminal points are nothing beyond the relation.* It is another superstition to fancy relations as an arbitrary network stuck on from the outside by destiny or chance, and making no reasonable difference to anything. And the root of both superstitions is the same. It is the refusal to recognize that * I am prepared to go a good deal beyond this.3 If occasion offered I should be ready to argue that you can not have a relation between points that are not different in quality. Not only, for instance, must spaces related be more than a mere relation in space, but they must also have a difference in quality. It is not possible to contemplate points in relation unless you distinguish them by a qualitative reference to the right or left or upper or lower sides of your body, and the different sensations which are at the root of these divisions, or again unless, by a qualitative mark such as A or B, you choose to make one different from the other. It may be objected that in certain cases the difference of quality is only one aspect of the whole relation. This view at least recognizes the existence of the difference, and I will not here discuss it. The ultimate connection of quality and relation is a most difficult problem. But it is clear that taken in their phenomenal appearance the one can not be reduced to the other. Is this double aspect true of the reality? Has that, as we are forced in the end to apprehend it, a single nature which combines two sides, and is so the root of the double appearance? Can we suppose that qualities are generated by the strife of some counterpart of what appears to us as relations? Or is it true that supersensible qualities are the reality which we perceive as phenomenal relations? Or is the question un answerable? If it is, we at least must not do violence to the given on the strength of a theory which we can not defend (cf. Book I. Chap.
2QO THE PRINCIPLES OF LOGIC BOOK II. PT. I the content of the given has always two sides,4 sensible quali ties and relations, and that one side can never, except by an artifice, be separate from or merged in the other. I do not say that these two elements are metaphysically irreducible; I do say that, taking them each as it stands, you must treat them each as a character of the given. It is a dire illusion to take the content of the given as either qualities without relation or relations without qualities, or to treat the one side as external to the other. Both are given together and given within the content. It was shown above (Bk. I. Chap. II. §21) that space and time-relations are no principium in- dividuationis; for they fall within the what, and do not make the this.
And another result was brought out in that Chapter. Un less judgments of sense make a false assertion they affirm or deny connections of content, and they do not affirm any thing else whatever. It is absurd to object that if Caesar is the same, he is in Gaul and in Italy, two places at once, or that if he is thirty he is also twenty-nine. The " at once " and the "also " conceal the old error. Of course it is not true that the identical Caesar under the same conditions 5 can be differently related to Italy in space or to his own birth in time; but then surely the conditions vary indefinitely. The mere lumping together unspecified conditions under the head " is now " does not show that the conditions are indiscernible, and that striking the differences out of the account we are forced to predicate contradictions of Caesar. What is true of Caesar in a certain context is true of the same Caesar in any other context. But this does not mean that one context is the other or is to be confused with the other. It means that Caesar has two different contexts, and that the truth of one can be no reason whatever for the falsehood of the other. If we fancy this is so we have given to one or to both assertions a meaning which is false, and we must be sent back once more to study the discussions of Book I. Chapter II.
§ 7. And there is another misunderstanding against which we must guard. That what is true of B here is true of B everywhere, means that, wherever B happens to be, you can say of it always what you have said of it once. This B you assert of is the self-same B that appears in the differences, but CHAP. VI TWO CONDITIONS OF INFERENCE 2QI it is not the B just as it appears in those differences. In A — B, B — C, the B is identical, and A and C are connected by that identity. But A and C are not themselves identical, and you can not predicate B — C of A — B. The B, of which what has once been said holds good for ever, is not the B which is one thing with A or one thing with C. It is the ab straction,6 the idealized content B, which is different from its contexts and yet connected with them, and on the strength of its oneness connects them together. The identity is always a synthesis of differences which themselves are not identical the one with the other, and apart from these differences the identity disappears into blank indiscriminateness.
I will try to illustrate the whole question briefly. We have a shed in the corner of a field, and, that shed being burnt, another is set up not distinguishable in itself from the first. Let the first be B — A and the second B — C; in what sense is it true that what holds of B once will hold of it always? The objection is obvious, In the shed B — A an event D hap pened, but can we say that the event took place in B — C? And if we can not say that, and if B is not distinguishable, how are we going to defend our axiom?
We are in no kind of perplexity. The content B is ob viously not the individual shed. The two sheds are made individual by their places in the series, and those places fall outside the abstraction B. What is true of B is universal propositions and is nothing besides. The event D can not be asserted truly until it becomes a hypothetical statement (Book I. Chap. II.).
But the objection will be pressed, " The sheds and their environment are a certain content, and that content is the same. If, on the strength of this content, we said of the shed B — A ' D happened here yesterday/ why can we not also upon this ground now say of the shed B — C ' D happened here last year'? The content is what we go from, and we have that in both cases." I reply, By all means: the content is the same. Let us try to carry out the process you recom mend. We can not of course connect D with B — C unless we establish a chain of relations through the identity of their end-points (ibid.). You can not go direct from the content. to the temporal event D, for that, as we have seen, is not THE PRINCIPLES OF LOGIC BOOK II. PT. I predicated categorically (ibid.).7 You must start from the content as given in one time. Well, starting from B — A you got a chain of events which took you back to D. But, if you start from B — C, you have a chain of events which takes you back first to the origin of B — C, when B did not exist, and then again through the destruction of B — A, to the time when B once more existed and was connected with D. Your process informs you that D the event will not fall within the identity of the ideal content B — C. That content has been qualified by a limitation in time, and qualified again by a definition of its component elements, which excludes their identity with the elements of B — A. If you deny that these qualifications are objects of knowledge, then I admit D is true of B — C, and why in the world should we not think it true? But if you admit that these qualifications are distinctions, then the content of the sheds is not indiscernible, and therefore by your admission is not identical.8 This, I think, is a sufficient answer to the objection, but it omits to take notice of several difficulties. There are ques tions which no doubt might occasion us trouble, but they do not seem to concern us here. We have been forced to notice a metaphysical problem which, at least in this work, we can not deal with, and hence objections which we can not here attempt to answer may be directed against us. But at least on one side I think we are safe; we need fear no col lision with the Philosophy of Experience, for that philosophy does not know the ground it stands on. Since Hume's bold speculations on the subject of identity were suppressed by himself, the English school has repeated a lesson by rote and flaunted a blind ancestral prejudice.
§ 8. The importance of the subject may excuse a repe tition. That what is the same ideally is really the same is without any doubt an enormous assumption, and I do not say that this assumption is true. What I do say is (a) that all inference presupposes it, and (b) that the objection to it rests on nothing but metaphysics.
(a) If we only will look at the palpable facts, we must admit that logic stands or falls with this axiom. Wherever we join one premise with another we must do so by means of an identical point, which, given as it is in diverse presenta- CHAP. VI TWO CONDITIONS OF INFERENCE 2Q3 tions, is held to be the same because it has the same content, and which, so far as it is not ideally discernible, is taken as one. Failing this identity the construction falls apart. I confess I do not know how to make this any clearer. I can only say to any one who doubts it, Show me an inference where this does not hold good, and I will show you a vicious inference, and you yourself shall admit that it is vicious.
(b) It sounds terrible to say that Identity is an ideal syn thesis of differences, and that this identity is real fact. The words are strange to the common mind, but it has always tacitly accepted their meaning. We believe that a body has changed its place, but at the end of the movement the change that is past is no fact of sense. We abstract the body from its present position and, treating this abstraction as a con tinuous identity, we predicate of it the changing differences. But do we doubt that motion is a real fact? And if we are told, It is the material atoms which are the same throughout; then why I would ask do we take them for the same, despite their differences of time and space, except because their ideal content is the same? The identity of indiscernibles may be true or false, but not only is it impossible to reason without it, but it is the abstract formula for our common-sense belief.
The authority of common sense is no authority for me, but the result we have reached may bring out one fact.^ The objection, raised by the Philosophy of Experience against a real identity, does not rest on any difficulty felt by common sense, and it is not an objection it would ever think of raising. It is a metaphysical objection, and it rests entirely on a metaphysical doctrine. It is because the Philosophy of Ex perience is sure that there is no reality except exclusive par ticulars, that it is horror-struck at the thought of a real universal. And because its belief is not proved nor thought to need proof, nor in any way discussed, because it is a mere inherited preconception which has got to think itself a real fact, it is scarcely so much to be called a doctrine as an orthodox dogma and traditional superstition.
And, as it must happen with all orthodox dogmas, its votaries do not take their professions in earnest. If an uni versal content may ever be real, on what ground can they deny the identity of thoughts because one is yesterday and 294 THE PRINCIPLES OF LOGIC BOOK II. Px. I the other to-day? But if such ideal sameness is not real, then how can any process or change or continuity be any thing but illusion? If a thing is not now the same that it was, if it is only alike, then it can not have changed. And if it is the same, on what ground do we make that assertion except on the ground of identity of content? It is frivolous9 to say that identity may be real, where existence is continuous and is not broken in the series of time, but is not real any where else. For if you allow that any lapse or change is a fact, you have admitted the reality of an element not confined to this or that particular, and you have admitted it on the ground of the identity of indiscernibles. You have already thrown your principle overboard, and if it is false in one place it may be false in another. Or to put the same thing in another form, if you are afraid to break with common sense in one point, what makes you so very bold in another? If I am to answer the question for you, I am forced to say that you have partly no head and partly no heart. You do not see the consequences deducible from your doctrine, and when a consequence begins to look like a reductio ad absurdum, you refuse to follow it. And this is what we call or used to call