I must remark, in the first place, that I wholly fail to see the absurdity. If you mean to exhaust the cases which ex- 134 THE PRINCIPLES OF LOGIC BOOK I elude the term " malleable-dense-metal," the absurdity would lie in their number being less than what follows from the number of possible combinations. But if you mean to say that, if "or " is exclusive, you can not deny the term which is offered unless you set out all the cases which exclude it, then this is just the mistake we have been considering. In " not- malleable or not-dense or not-metallic" the disjoined are in compatible, but the full possibilities are not set out. You must understand with each " or " the implication of " failing that." " Not malleable " does not mean the isolated presence of non-malleability. It is not one possibility: it is a class that covers several. It means the absence of malleability, whether the subject is metallic or not-metallic, dense or not- dense. You may fairly object that combinations are ignored, or else that the term " not-malleable " is ambiguous, since it is used to cover a number of cases. But these technical objec tions would have little importance, and they do nothing to show that " or " does anything but rigidly disjoin.
§ ii. Despite my respect for Professor Jevons, I can not admit any possible instance in which alternatives are not exclusive. I confess I should despair of human language, if such distinctions as separate " and " from " or " could be broken down. And, when I examine the further evidence produced, it either turns on the inaccurate modes of expres sion we have lately discussed, or consists in what I must be allowed to call a most simple confusion. We are told that the expressions " wreath or anadem," or again " unstain'd by gold or fee " (Jevons, p. 70), show that " or " may sometimes be non-exclusive. But this is quite erroneous. The alterna tives are meant to be rigidly incompatible. The distinction is however not applied to the thing, but simply to the names. If we suppose that the terms are quite synonymous, then " wreath or anadem" means "you may call it by either name you please." The thing has two titles, one of which is at your service. I hardly think Professor Jevons would assert that we are asked to use both names at once. So, if " fee " is not meant to be distinct from " gold," the assertion is that there is no stain arising from the thing you may term indifferently gold or fee. The idea of your wanting to say both at once is quite ignored.
CHAP. IV THE DISJUNCTIVE JUDGMENT 135 I will try to make the matter clearer by inventing a piece of imaginary dialogue. A. Who is the greatest Roman poet? B. His name is Virgil. A. What, not Vergilius? B. Yes, Virgil or Vergilius. A. I understand: he has two names. I will call him henceforth " Vergilius- Virgil," and then I shall be safe. B. Excuse me: in that case you must be wrong. You may call him by either of the names you please, but not by both of them at once.
It is not worth while to multiply illustrations. In every instance that can be produced, we have either a loose mode of common speech, or else the " or " denotes incompatibility, whether that lie in the simultaneous use of alternative names, or in the facts themselves.
§ 12. The mere statement, of course, may fail to tell us which of these incompatibilities is before us. And no one can deny that alternatives are often presented in a very inac curate way. It is an excellent thing in all these questions to refer to the common usages of language, but we must remember that in those usages, besides what one calls " un conscious logic," there also may lurk mere looseness and care lessness. It may not be amiss to illustrate the mistake we have just been discussing, by a parallel ambiguity in the hypothetical judgment. It is, of course, the established doc trine that, while you may argue from ground to consequence, you can not demonstrate from consequence to ground.10 And, although from a metaphysical point of view this doctrine is certainly open to doubt, still for logical purposes it is suffi ciently valid. But yet, by appealing to loose expressions, we might show that the ground is the only ground, and can therefore be inferred from the presence of the consequence. Sigwart has called attention to these cases (Logik, I. 243; and Beitrage, 59). "If you run hard you will catch him," is often an indirect way of saying, " You will not catch him unless you run hard." But such mere loose phrases are no valid reason for impugning the doctrine that, unless this fact is specially stated, the condition is not given as a sine qua non. When the context shows that our expressions are not to be strictly interpreted, we are at liberty to take " either—or " as compatible, and " if " may be the same as " not unless." But 136 THE PRINCIPLES OF LOGIC BOOK I we should remember that what a thing can pass for may differ widely from what it really is.
§ 13. It is time we left these misleading errors to return to the discussion of the matter itself. The detail of the process in disjunctive judgments can not fully be dealt with till we come to inference. But here we may partly prepare the ground.
In the first place, as we shall see in the following chapter, disjunction does not rest on Excluded Middle. The latter is merely a case of disjunction.
" A is b or c " asserts, as we saw, that A exists and pos sesses a quality. That quality, further, falls within be. It is affirmed to be what is common to both, and it is stated also to be further determinable within be. In other words, it excludes all discrepant with both b and c.
We have seen this above, and the point I wish here to bring forward is the following. How do we know, and how can we know, that there is not something discrepant with be and yet compatible with A? All rests upon this; and what does this rest on?
We must answer, for the present, that it rests on our impotence.11 There is no great principle on which we can stand. We can not find any opposite of b or opposite of c which is not also an opposite of A; and we boldly assume that, because we find none, therefore there is none. The con clusion from impotence may itself seem impotent, but, as we shall hereafter see, there remains some doubt if it may not in the end be taken as the ground and the sole ground we have for believing anything (Book III. II. Chapter III.).
§ 14. We may state the whole matter once more thus. " A is b or c" may be expressed by (i) If A is b it is not c, and If A is c it is not b, (ii) If A is not b then it is c, and If A is not c then it must be b. The first two hypothetical state ments are erected on the knowledge that b and c as predicates of A are incompatible, or that Abe can not possibly exist.
The second pair are based on the assumption that, because we do not find a predicate of A which excludes b or c, there fore there is none. Every opposite of b or of c, that we find, is an opposite of A. Hence there remains this result; within the limit of A there is no not-fc but c, and no not-c but b: and CHAP. IV THE DISJUNCTIVE JUDGMENT 137 A must have some further quality. This is the ground for our second two hypotheticals.
So we see the essence of disjunctive judgment is not got by calling it a combination of supposals. It has a distinctive character of its own. It first takes a predicate known within limits, and defined by exclusion, and then further defines it by hypothetical exclusion. It rests on the assumption that we have the whole field, and by removing parts can determine the residue. It supposes in short a kind of omniscience. Its assertion again, if not quite categorical, is certainly not quite hypothetical. It involves both these elements. And it implies, in addition, a process of inference which will give us cause for reflection in the future.
ADDITIONAL NOTES ADDITIONAL NOTES 1 On the subject of Disjunction the reader is referred to Dr. Bosanquet's Logic. I fully accept his main view; but, before pro ceeding in consequence to point out some errors made in this volume, I will add a few remarks which may perhaps assist the reader.
Disjunction means "or," and, viewed psychologically, "or" stands for Choice. Hence it may be useful to consider here how choice arises. Where something is desired, where there are various ways of realizing this end, and where I find that I can not have all of these as a whole and at once — and where, by this negation, action has been suspended — the result may be choice. And, in choosing, I accept one way while rejecting the rest. Or, again, I foresee, let us say, that an event affecting me must actually happen, and that, so far, there are no two ways, but that, as to how the event will happen and affect me, the ways are various. And let us add that I perceive that these diverse modifications, while impossible all at once, are otherwise, each of them, more or less in my power. Here, as above, after suspend ing action I identify myself in choice with one of these modes, while rejecting the rest. And obviously the "or" thus contained in choice is exclusive; and any other view as to "or" would, here at least, conflict with plain fact.
We may note further that in choice the alternative need not merely be dual. The incompatibles that are each possible may be clearly more than two. And this plurality in the " or " holds, I would add, not merely psychologically and in choice, but equally belongs to the "or" of logical disjunction. The necessary duality of disjunction —in the sense that the incompatibles, which are each possible, can not be taken as more than two — is to my mind a view which, so far, is contrary to fact.
THE PRINCIPLES OF LOGIC BOOK I What is true here is that when, and so far as, in choice or other wise, you identify yourself with one possibility, the residue tends to be regarded or at least treated, so far, as not even possible. It is taken for our purpose, we may say, in the lump and as all one, and so, we may add, is taken but as one only. But in the above attitude (to speak strictly) there is really no question of number, since what is rejected is viewed merely from the side of its general quality.
Having now noticed the character of " or " as we find it in choice, I will go on to deal briefly with the origin and nature of logical dis junction. Before this arises we have found that Reality, as this or that recognizable object, has various qualities. We have passed, that is, beyond the stage of an immediate unity of one and many. We have reached a level where the distinguished many are recognized as diverse, and yet, as each and also the rest, belonging somehow to the one object. And the object possesses this diversity, so far, all together and at once. The qualities thus seem simply conjoined and are called compatible. But we go on to discover, in the same or in some other object, qualities not found thus together, and we call these, so far, incompatible. The object may have now one and now another, but never has at once both or all. On the other side (here is the point) the object has these qualities. It has them, now one and now another, according to the conditions. It has them, that is, not simply, but according as it itself is made diverse, so as itself to enter into and become this one qualification while rejecting the others. And, since we find that this holds good also with regard to these other incompatibles, the object is now qualified by them all, but qualified disjunctively. While on one hand the object is all, it is on the other hand each singly and each exclusively according to the diverse con ditions.
Such in outline, we may say, is the origin of Disjunction, and its intellectual importance and necessity can not well be overestimated. But we are led none the less to look beyond it to a higher and more ultimate stage, where we return to Conjunction in a different sense and at a higher level. In a complete and perfect system, where all conditions were filled in, the real Universe would have all its de terminations at once, all as connected and each as qualifying the others and the whole. And here negation would disappear except as one aspect of positive and complementary distinction. But for us this ultimate stage of the intellect remains an ideal, in the sense that it can not in detail and everywhere be attained completely.
This ideal, I must add, in no way justifies the doctrine that in logic " or " can anywhere have a sense which is not exclusive. In other words — so far as " or " ceases to exclude — both " or," and disjunction with it, have so far ceased as such to exist. See below, 2 " To state it incorrectly." I can not now recall the origin of this remark, which seems at best negligible. For the possibility of reduc tion to two hypothetical see Bosanquet, K & R, p. 208.
CHAP. IV THE DISJUNCTIVE JUDGMENT 139 31 " It is nonsense." Certainly (I must insist) it is so. We may illustrate by the attempt to reduce relational wholes, and even per haps all facts, to relations and terms.
4 "The basis...is categorical." This, however, is subject to a qualification, for which see Bosanquet, Logic (Ed. II), I, 328.
5 It is of course not true that disjunction assumes the existence of its subject (cf. Chap. V, §23), if that means "existence in my real world." The subject may be hypothetical or otherwise "imaginary." But there is in the end an Ultimate within which all disjunction falls and which has no negative.
6 The use of "quality" here is objectionable. See on Chap. II, §50.
7 " It can not be both and it must be some one." If " some one " means " one only," this would anticipate the disjunction. It should mean " must be, so far, within the field of both while not both." The disjunction implied in "one only" requires hypothetical to complete it and make it explicit. The same requirement, though not in the same sense, holds, I think, of our knowledge that b and c are in compatible.
8 I am more than ever convinced that the view which takes " or " as not always exclusive is utterly untenable, except perhaps by way of of a mere convenient artifice. On the whole question, and for a refuta tion of technical objections, see Bosanquet, Logic (Ed. II), Vol. I, pp. 355 foil. The subject is now perhaps exhausted, but I will allow myself to add one or two remarks.
(i) The evidence from psychology seems to me to be all on one side. " Or " answers to choice, and choice seems nonsense if it means that you can have all at once. And this consideration ought, I submit, to carry great weight.
(ii) The fallacy of " false alternative," I would further remind the reader, may be said perhaps to dominate our lives. But how could this be so, if "or" were not taken instinctively as everywhere exclu sive? And the fallacy does not lie in our assuming this as true. It lies, on the other hand, in our forgetting constantly the actual nature of our subject. We may say that (apart from exceptional cases) in every judgment made in life the real subject is other than that which is formulated. It is really that which is "understood" for the purpose in hand, which limited purpose, not being made explicit, is easily ignored or forgotten. Thus, in " A is b or c," the A which we mean is A qualified and limited by our special object and interest. It is of and within this qualified A that our " b or c " holds. And it is when we ignore or forget this, and when we go on to take A simply and unrestrictedly in the sense of " A anyhow," that the fallacy everywhere tends to arise. But this tendency, with its false result, points, I submit, to our unfailing reliance on the exclusiveness of "or."
The arguments against its exclusiveness seem to me plausible only when this unexpressed qualification of A is ignored. In our actual use of " or," A means an A not only where the possibility of Abe is excluded, but also where it is tacitly set aside as irrelevant to our I4O THE PRINCIPLES OF LOGIC BOOK I purpose. And, when we remember this, the case against exclusiveness, to my mind, disappears.
(iii) If we keep to mere "or," then "or" is both exhaustive and exclusive; and, if we are to argue against the second of these characters, we must also argue against the first. After stating that A is & or c, we may receive the answer "yes or both" without any feeling that our statement has been denied. But, on just the same principle, we may accept a suggested addition of "or d" not as a correction but as a complement. And yet to argue from this that disjunction is not taken as exhaustive would, if plausible, be erroneous. The " or d " is ac cepted because, and so far as, A did not mean "A pure and simple." It in fact meant, for our purpose, something like " A, whatever else A elsewhere may be." Hence a correction and a replacement of A by a wider subject is (we feel) not called for here by the addition of "or d." And thus the actual disjunction is really exhaustive, just as on the same principle (we saw above) it was really exclusive — unless, that is, the disjunction in both cases has ceased, as such, to exist, and has really given way to some lower or higher mode of assertion.
9 " His statement is...false." And it therefore, I presume, can lead formally to the equally false result " He is either wise or honest " (Keynes, Formal Logic, p. 280, note). This, if correct, I take to confirm the doctrine of my text, which doctrine, I venture to think, here and elsewhere, Dr. Keynes has failed to understand.
10 "The established doctrine." The doctrine is, however, in the end untenable. See Bosanquet, Logic I, Chap. VI. I have fully accepted his view; see Appearance, Index, s. v. Cause, and T. E. X. Apart from this, the instance in the text may serve to illustrate the ambiguity of ordinary language.
11 For privation and impotence as a ground of knowledge see on Chap. Ill, §9. So far as our knowledge is completely systematized, privation, of course, so far ceases to exist.
CHAPTER V THE PRINCIPLES OF IDENTITY, CONTRADICTION, EXCLUDED MIDDLE, AND DOUBLE NEGATION.
§ i. After discussing negative and disjunctive judgments, we may deal at once with the so-called " Principles " of Iden tity, Contradiction, and Excluded Middle; and we will add some remarks on Double Negation.
The principle of Identity is often stated in the form of a tautology, " A is A." If this really means that no difference exists on the two sides of the judgment, we may dismiss it at once. It is no judgment at all. As Hegel tells us, it sins against the very form of judgment; for, while professing to say something, it really says nothing. It does not even assert identity. For identity without difference is nothing at all. It takes two to make the same, and the least we can have is some change of event in a self -same thing, or the return to that thing from some suggested difference. For, otherwise, to say " It is the same as itself " would be quite unmeaning. We could not even have the appearance of judgment in " A is A," if we had not at least the difference of position in the different A's; and we can not have the reality of judgment, unless some difference actually enters into the content of what we assert.
§ 2. We never at any time wish to use tautologies. No one is so foolish in ordinary life as to try to assert without some difference. We say indeed " I am myself," and " Man is man and master of his fate." But such sayings as these are no tautologies.1 They emphasize an attribute of the subject which some consideration, or passing change, may have threatened to obscure; and to understand them rightly we must always supply " for all that," " notwithstanding," or again, " once more." It is a mere mistake to confuse what Kant calls " analytical judgments " * with tautologous state ments. In the former the predicate is part of the content of the conception A, which stands in the place of, and appears as, *This is not the sense in which I have used "analytical." p. 48.
42 THE PRINCIPLES OF LOGIC BOOK I the subject. But in every judgment of every kind a synthesis is asserted. The synthesis in Kant's analytical judgment holds good within the sphere of the conception; and the real subject is not the whole of A, but is certain other attributes of A which are not the attribute asserted in the predicate. In " All bodies are extended " what we mean to assert is the connection, within the subject "bodies," of extension with some other property of bodies. And even if " extended " and " body " were synonymous, we still might be very far from tautology. As against some incompatible suggestion, we might mean to assert that, after all misapprehension and improper treatment, the extended is none the less the extended. And, again, we might be making a real assertion of a verbal nature. We might mean that, despite their difference as words, the mean ing of " body " and " extended " was the same. But mere tautology with deliberate purpose we never commit. Every judgment is essentially synthetical.
§ 3. The axiom of Identity, if we take it in the sense of a principle of tautology, is no more than the explicit statement of an error. And the question is, would it not be better to banish irrevocably from the field of logic such a source of mistake? If the axiom of Identity is not just as much an axiom of Difference, then, whatever shape we like to give it, it is not a principle of analytical judgments or of any other judgments at all. On the other hand, perhaps something may be gained if a traditional form can get a meaning which con veys vital truth. Let us try to interpret the principle of Iden tity in such a way that it may really be an axiom.
§ 4. We might take it to mean that in every judgment we assert the identity of subject and predicate. Every connection of elements we affirm, in short all relations and every differ ence, holds good only within a whole of fact.2 All attributes imply the identity of a subject. And taken in this sense the principle of Identity would certainly be true. But this perhaps is not the meaning which, for logical purposes, it is best to mark specially.
§ 5. There remains a most important principle which, whether it be true or open to criticism, is at least the sine qua non of inference. And we can not do better than give this the name of principle of Identity, since its essence is to emphasize CHAP. V THE PRINCIPLE OF IDENTITY 143 sameness in despite of difference. What is this principle? It runs thus: " Truth is at all times true," or, " Once true always true, once false always false. Truth is not only independent of me, but it does not depend upon change and chance. No alteration in space or time, no possible difference of any event or context, can make truth falsehood. If that which I say is really true, then it stands for ever."
So stated the principle is not very clear, but perhaps it will find acceptance with most readers. What it means, how ever, is much more definite, and will be much less welcome. The real axiom of Identity is this: What is true in one context is true in another.5 Or, If any truth is stated so that a change in events will make it false, then it is not a genuine truth at all.
§ 6. To most readers this axiom, I have little doubt, will seem a false statement. For the present it may stand to serve as a test if our previous discussions (Chap. II.) have been understood. If every judgment in the end is hypothetical, except those not directly concerned with phenomena — if each merely asserts a connection of adjectives, in this sense that given A then B must follow — we see at once that under any conditions it will always be true. And we shall see here after that in every inference this result is assumed as a prin ciple of reasoning, and that we can not argue one step with out it.
§ 7. We saw that such judgments as " I have a toothache," in their sensuous form, are not really true. They fail and come short of categorical truth, and they hardly have attained to hypothetical. To make them true we should have to give the conditions of the toothache, in such a way that the con nection would hold beyond the present case. When the judg ment gave the toothache as the consequent coming according to law from the ground, when the judgment had thus become universal, and, becoming this, had become hypothetical, then at last it would be really true, and its truth would be uncondi tional and eternal.
I know how absurd such a statement sounds. It is impos sible, I admit, however much we believe it, not to find it in a certain respect ridiculous. That I do not complain of, for it is not our fault. But it is our fault if the common view does not seem more ridiculous. I say that " I have a toothache " 144 THE PRINCIPLES OF LOGIC BOOK I to-day. It is gone to-morrow. Has my former judgment become therefore false? The popular view would loudly pro test that it still is true, for I had a toothache, and the judg ment now holds good of the past. But what that comes to is simply this. The judgment is true because answering to fact. The fact alters so that it does not answer; and yet the judg ment is still called true, because of something that does not exist. Can anything be more inconsistent and absurd? If the change of circumstance and change of day is not a fresh con text which falsifies this truth, why should any change of context falsify any truth? And if changed conditions make any truth false, why should not all truth be in perpetual flux, and be true or false with the fashion of the moment?
§ 8. We shall discuss this question more fully hereafter (Bk. II. Part I.), but may here anticipate a misunderstanding. To ask " Does space or time make no difference " is wholly to ignore the meaning of our principle. We ask in reply, " Does this difference enter into the content of A? If it does, then A becomes perceptibly diverse, and we confessedly have left the sphere of our principle. But, if it does not so enter, then the truth of A is considered in abstraction from spaces and times, and their differences are confessedly irrelevant to its truth. We thus meet the objection by offering a dilemma. You have abstracted from the differences of space and time, or you have not done so. In the latter case your subject itself is different; in the former case it is you yourself who have excluded the difference.
We may indeed on the other side be assailed with an ob jection. We may be asked, "What now has become of the identity? Has it not disappeared together with the differ ences? For if the different contexts are not allowed to enter into the subject, how then can we say what is true in one con text is true in another? It will not be true in any context at all." But we answer, The identity is not contained in the judg ment " S — P," since that takes no kind of account of the dif ferences.4 The identity lies in the judgment, " S — P is true everywhere and always." It is this "everywhere" and "al ways " that supply the difference against which S — P becomes an identity. The predicate attributed to the real belongs to it despite the difference of its diverse appearances. We do not say CHAP. V THE PRINCIPLE OF CONTRADICTION 145 the appearances are always the same, but the quality keeps its nature throughout the appearances. And with this reply we must here content ourselves.
§ 9. When we come to discuss the nature of inference we shall see more fully the bearing of the principle. It stands here on the result of our former enquiries, that every judg ment, if it really is true, asserts some quality of that ultimate real which is not altered by the flux of events. This is not the place for metaphysical discussion, or we might be tempted to ask if identity was not implied in our view of the real. For if anything is individual it is self -same throughout, and in all diversity must maintain its character.
THE PRINCIPLE OF CONTRADICTION.
§ 10. Like the principle of Identity, the principle of Con tradiction has been often misunderstood. And in the end it must always touch on a field of metaphysical debate. But, for logical purposes, I think it is easy to formulate it in a satisfac tory way.