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.
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.
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 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.
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 " 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.
It is necessary before all things to bear in mind that the axiom does not in any way explain, that it can not and must not attempt to account for the existence of opposites.5 That discrepants or incompatibles or contraries exist, is the fact it is based on. It takes for granted the nature of things in which certain elements are exclusive of others, and it gives not the smallest reason for the world being such in nature and not quite otherwise. If we ever forget this, the Law of Contra diction will become a copious source of illusion.
§ II. If the principle of Contradiction states a fact, it says no more than that the discrepant is discrepant, that the exclu sive, despite all attempts to persuade it, remains incompatible. Again, if we take it as laying down a rule, all it says is, " Do not try to combine in thought what is really contrary. When you add any quality to any subject, do not treat the subject as if it were not altered. When you add a quality, which not only removes the subject as it was, but removes it altogether, then do not treat it as if it remained." This is all the meaning it is safe to give to the axiom of Contradiction; and this meaning, I think, will at once be clear, if we bear in mind our former discussions. The contrary is always the base of the contradictory, and the latter is the general idea of the contrary. Not-A for example is any and every possible contrary of A § 12. We have to avoid, in dealing with Contradiction, the same mistake that we found had obscured the nature of Iden tity. We there were told to produce tautologies, and here we are by certain persons forbidden to produce anything else. " A is not not-A " may be taken to mean that A can be nothing but what is simply A. This is, once again, the erroneous asser tion of mere abstract identity without any difference. It is ordering us to deny as a quality of A everything that is differ ent from A, and in this sense not-A. But differents and discrepants should never be confused. The former do not exclude one another; they only exclude the denial of their difference. The discrepant with A can never be found together with A in any possible subject, or be joined to it in the relation of subject and attribute.6 The different from A does not exclude, unless you attempt to identify it with A. It is not A generally, but one single relation to A, which it repels.
As we saw before, there is no logical principle which will tell us what qualities are really discrepant. Metaphysics, in deed, must ask itself the question if any further account can be given of incompatibility. It must recognize the problem, if it can not solve it. We might remark that no thing excludes any other so long as they are able to remain side by side, that incompatibility begins when you occupy the same area; and we might be tempted to conclude that in space would be found the key of our puzzle. But such other experiences as that assertion and denial, or pain and pleasure, are incompatible, would soon force us to see that our explanation is insufficient. But in logic we are not called upon to discuss the principle, but rest upon the fact. Certain elements we find are incom patible; and, where they are so, we must treat them as such.
§ 13. There is no real question of principle involved in such different ways of stating the axiom as " A is not not-A," " A is not both b and not-b" " A can not at once both be and not be." For if A were not-A, it would be so because it had some quality contrary to A. So also, if A has a quality b, it could only be not-b by virtue of a quality discrepant with b. And again, if A both were and were not, that would be be- CHAP. V THE PRINCIPLE OF CONTRADICTION 147 cause the ultimate reality had contrary qualities. The charac ter in which it accepted A, would be opposite to the quality which excluded A from existence. Under varieties of detail we find the same basis, repulsion of discrepants.
A simple method of stating the principle is to say, " Denial and affirmation of the self -same judgment is wholly inad missible." And this does not mean that if a miracle in psy chology were brought about, and the mind did judge both affirmatively and negatively, both judgments might be true. It means that, if at once you affirm and deny, you must be speaking falsely. For denial asserts the positive contrary of affirmation.7 In the nature of things (this is what it all comes to) there are certain elements which either can not be con joined at all, or can not be conjoined in some special way; and the nature of things must be respected by logic.
§ 14. If we wish to show that our axiom is only the other side of the Law of Identity, we may state it thus, " Truth is unchangeable, and, as discrepant assertions alter one another, they can not be true." And again, if we desire to glance in passing at the metaphysical side of the matter, we may remind ourselves that the real is individual, and the individual is har monious and self-consistent. It does not fly apart, as it would if its qualities were internally discrepant.
§ 15. Having now said all that I desire to say, I would gladly pass on. For, notwithstanding the metaphysics into which we have dipped, I am anxious to keep logic, so far as is possible, clear of first principles. But in the present instance the law of Contradiction has had the misfortune to be flatly denied from 8 a certain theory of the nature of things. So far is that law (it has been contended) from being the truth, that in the nature of things contradiction exists. It is the fact that opposites are conjoined, and they are to be found as discrepant moments of a single identity.
I need hardly say that it is not my intention compendiously to dispose in a single paragraph of a system which, with all its shortcomings, has been worked over as wide an area of ex perience as any system offered in its place. My one idea here is to disarm opposition to the axiom of contradiction, as it stands above.9 But I clearly recognize that, if not-A were taken as a pure negation, no compromise would be possible. You would then have to choose between the axiom of contra diction and the dialectical method.
I will say, in the first place, that whatever is conjoined is therefore ipso facto shown not to be discrepant. If the ele ments co-exist, cadit qucestio; there is no contradiction, for there can be no contraries. And, saying so much, I feel tempted to retire. But yet with so much I shall hardly escape. " Have not we got," I hear the words called after me, " have we not got elements which any one can see negate one another, so that, while one is, the other can not be; and yet have we not got very many conceptions in which these discrepants somehow co-exist? It is all very well to say, * then not con trary; ' but try them, and see if they are not exclusive."
It is plain that I must stand and say something in reply. But I think I shall hardly be so foolish as to answer, " These conceptions of yours are merely phenomenal. Come to us and learn that knowledge is relative, and with us give up the Thing-in-itself." For without knowing all that would be poured on my head, I can guess some part of what I should provoke. " You say ' give up the Thing-in-itself '? Why that is all that you have not given up. You profess that your knowledge is only phenomenal, and then you make the law of Contradiction valid of the Absolute, so that what it excludes you are able to know is not the Absolute. That is surely in consistent. And then, for the sake of saving from contradic tion this wretched ghost of a Thing-in-itself, you are ready to plunge the whole world of phenomena, everything you know or can know, into utter confusion. You are willing to turn every fact into nonsense, so long as this Thing-in-itself is saved. It is plain, then, for which you really care most. And as for * relativity/ it is you yourselves who violate that prin ciple. Your turning of the relative into hard and fast con traries is just what has brought you to your miserable pass." I confess I should hardly care to subject myself to all these insults; and I had rather Mr. Spencer, or some other great authority — whoever may feel himself able to bear them, or unable to understand them — should take them on himself.
If I chose to turn and provoke a contest, I know of another weapon I might use. I might say, " Your conceptions CHAP. V THE PRINCIPLE OF CONTRADICTION 149 4are partial illusions. They are crude popular modes of repre senting a reality whose nature can not be so portrayed. And the business of philosophy is to purify these ideas, and never to leave them until, by removal of their contradictions, they are made quite adequate to the actual fact." But, after all, perhaps I could only say this for the sake of controversy, and controversy is what I am anxious to avoid. And for this end I think that some compromise may perhaps be come to. With out calling in question the reality of negation, and the identity of opposites, are we sure that we can not understand that doctrine in a sense which will bear with the axiom of Contra diction? This axiom is not like the principle of Identity. It is a very old and most harmless veteran; and for myself I should never have the heart to attack it, unless with a view to astonish common-sense and petrify my enemies. And in metaphysics we can always do that in many other ways.
What I mean is this.10 Supposing that, in such a case as continuity, we seem to find contradictions united, and A to be b and not-b at once, this may yet be reconciled with the axiom of Contradiction. A we say is composed of b and not-b; for, dissecting A, we arrive at these elements, and, uniting these, we get A once more. But the question is, while these elements are in A, can they be said, while there, to exist in their fully discrepant character of b and not-b? I do not mean to sug gest that the union of contraries may be that misunderstanding of the fact which is our only way to understand it. For, if I felt sure myself that this were true, I know it is a heresy too painful to be borne. But, in the object and within the whole, the truth may be that we never really do have these discrepants. We only have moments which would be incompatible if they really were separate, but, conjoined together, have been subdued into something within the character of the whole. If we so can understand the identity of opposites — and I am not sure that we may not do so — then the law of Contradiction flourishes untouched. If, in coming into one, the contraries as such no longer exist, then where is the contradiction?
But, I fear, I shall be told that the struggle of negatives is the soul of the world, and that it is precisely because of their identity that we have their contradiction. It is true that the opposition which for ever breaks out leads to higher unity in