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 146 THE PRINCIPLES OF LOGIC BOOK I 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 dis- crepants 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 148 THE PRINCIPLES OF LOGIC BOOK I 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 4- are 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 discrep- ants. 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 I5O THE PRINCIPLES OF LOGIC BOOK I which it is resolved; but still the process of negation is there. It is one side of the world which can not be got rid of, and it is irreconcileable with the non-existence of discrepants in a single subject. Each element of the whole, without the other, is incompatible with itself; but it is none the less incompatible with the other, which for ever it produces or rather becomes.
I am after all not quite convinced. If the law of Contra diction is objected against because, in isolating and fixing the discrepant, it becomes one-sided, is it not quite possible that, in denying the law, we have become one-sided in another way? If the negation itself, while negative on one side, is on the other side the return from itself to a higher harmony — if, that is to say, the elements are not discrepant without each at once, by virtue of its discrepancy and so far as it is discrepant, thereby ipso facto ceasing to be discrepant, then surely, in denying the law of Contradiction, we ourselves have fixed one side of the process, and have treated the contrary as simply contrary. The contrary which the law has got in its head, is the contrary that entirely kills its opposite, and remains tri umphant on the field of battle. It is not the contrary whose blows are suicidal, and whose defeat must always be the doom of its adversary. It is incompatibles fixed as such, it is dis crepants which wholly exclude one another and have no other side, that the axiom speaks of. But dialectical contraries are only partially contrary and it is our mistake if we keep back the other side. And if an opponent of the law reminds me that the existence of these two sides within one element is just the contradiction, that in the b which is contrary to not-& the implication of not-b makes it self-contradictory, then I must be allowed to say in reply that I think my objector has not learnt his lesson. The not-& in b is itself self-discrepant, and is just as much b: and so on for ever. We never have a mere one-sided contrary.
But it is one-sided and stationary contraries that the axiom contemplates. It says that they are found,11 and no sober man could contend that they are not found. No one ever did maintain that the dialectical implication of opposites could be set going in the case of every conjunction that we deny. It can hardly be maintained that there are no discrepants, except these contraries which at the same time imply each CHAP. V THE PRINCIPLE OF EXCLUDED MIDDLE 151 other. And the law of Contradiction does not say any more than that, when such sheer incompatibles are found, we must not conjoin them.
Its claims, if we consider them, are so absurdly feeble, it is itself so weak and perfectly inoffensive, that it can not quarrel, for it has not a tooth with which to bite any one. The controversy, first as to our actual ability to think in the way recommended by Hegel, and secondly as to the extent to which his dialectic is found in fact, can not only not be settled by an appeal to the axiom, but falls entirely outside its sphere. Starting from the fact of the absolute refusal of certain ele ments to come together, and wholly dependent upon that fact, so soon as these elements do come together the axiom ceases forthwith to be applicable. It is based upon the self-con sistency of the real, but it has no right to represent that con sistency except as against one kind of discrepancy. So that, if we conclude that the dialectic of the real would in the end destroy its unity, that has nothing to do with the axiom of Contradiction. Like every other question of the kind, the validity of dialectic is a question of fact, to be discussed and settled upon its own merits, and not by an appeal to so- called " principles." And I think I may venture to hazard the remark, that one must not first take up from uncritical views certain elements in the form of incompatible discrepants, and then, because we find they are conjoined, fling out against the laws of Contradiction and Excluded Middle. They, such as they are, can be no one's enemy; and since no one in the end can perhaps disbelieve in them, it is better on all ac counts to let them alone.
PRINCIPLE OF EXCLUDED MIDDLE.12 § 1 6. The axiom that every possible judgment must be true or false,13 we shall see is based on what may be called a principle. It is however doubtful if the axiom itself should receive that title, since it comes under the head of disjunctive judgment. We must not imagine that our axiom supplies the principle of disjunction. It is merely one instance and applica tion of that principle.
§ 17. If we recall the character of the disjunctive judg ment, we shall remember that there we had a real, known to 152 THE PRINCIPLES OF LOGIC BOOK I be further determined. Its quality fell (i) within a certain area; and (ii) since that area was a region of discrepants, the real was determined as one single member. On this basis1* we erected our hypotheticals, and so the " either — or " was completed.
Excluded Middle shows all these characteristics. In it we affirm (i) that any subject A, when the relation to any quality is suggested, is determined at once with respect to that predicate within the area of position and negation, and by no relation which is incompatible with both. And (ii) we assert that, within this area, the subject is qualified as one single member. And then we proceed to our " either — or."
§ 18. Excluded Middle is one case of disjunction: it can not be considered co-extensive with it. Its dual and con tradictory alternative rests on the existence of contrary opposites. The existence of exclusives without reference to their number is the ground of disjunction, and the special case of assertion and denial is developed from that basis in the way in which contradiction is developed from exclusion. Common discrepant disjunction is the base, and the dual alternative of b and not-b rests entirely upon this.
§ 19. Excluded Middle is one kind of disjunction: and we must proceed to investigate the nature of that kind, (i) Dis junction asserts a common quality. In "b or not-b" the common quality asserted of A is that of general relation to b. (ii) Disjunction asserts an area of incompatibles. Affirmation or denial of b is here the area within which A falls. The evidence that it does not fall outside and that all the dis crepants are completely given, may be called my impotence to find any other.15 (iii) Disjunction attributes to the subject A one single element of the area. And this part of the process does not call here for any special remark.
§ 20. We find however, when we investigate further, a point in which the axiom of Excluded Middle goes beyond the limits of disjunctive judgment. It contains a further principle, since it asserts a common quality of all possible existence. It says, Every real has got a character which determines it in judgment with reference to every possible predicate. That character furnishes the ground of some judgment in respect of every suggested relation to every CHAP. V THE PRINCIPLE OF EXCLUDED MIDDLE 153 object. Or, to put the same more generally still, Every element of the Cosmos possesses a quality, which can determine it logically in relation to every other element.
§21. This principle is prior to the actual disjunction. It says beforehand that there is a ground of relation, though it does not know what the relation is. The disjunction proceeds from the further result that the relation falls within a dis crepant sphere. We thus see that, on the one hand, Excluded Middle transcends disjunction, since it possesses a self-de termining principle which disjunction has not got. On the other hand, in its further development, it is nothing what ever but a case of disjunction, and must wait for the sphere of discrepant predicates to be given it as a fact.16 § 22. The disjunction is completed by the fact that, when any predicate is suggested, the quality of every element is a ground of either the affirmation or the denial of the predicate. It compels us to one and to one alone; for no other alternative can possibly be found.
And here the opposition, directed before against the axiom of Contradiction, must again be confronted. It is false, we are told, that A must either be c or not-c. We have often to say " both," and sometimes " neither." But I think perhaps the discussion at the end of the foregoing chapter will have strengthened us to persist. I fully admit that often, when challenged to reply Yes or No, it is necessary to answer " Yes and No" or "Neither." But, I venture to think, that is always because the question is ambiguous, and is asked from the standpoint of a false alternative.17 " Is motion continuous? Yes or no." I decline to answer until you tell me if, by saying Yes, I am taken to deny that it is also discrete. In that case perhaps, instead of saying Yes, I should go so far as to answer No. There may be a middle between continuity and discretion; there can be none between continuous and not- continuous.
The ground of the objection to the Excluded Middle is, I am bold enough to think, fallacious. Given not fixed dis- crepants but dialectical opposites, the existence of these together in one single subject does not give us the right to a negative judgment. One can not be made use of as the positive ground on which to build the denial of the other.
THE PRINCIPLES OF LOGIC BOOK I One does not wholly remove the other, and, failing to do so, it is not qualified as a logical contrary. For it is only the discrepant which destroys its opposite that can serve as the base of a negative judgment. And, failing the denial of one quality through the other, the answer must be that both are present, and the denial of either is wholly excluded. But I fear it is hard altogether on this point to effect a compromise. If the negative of b is ever simply not-&, and if this is the other which is implicated with b in one subject A, then I grant the Excluded Middle disappears. But, I think, in this case it will carry along with it enough to ruin what is left behind. And I must leave the matter so.
§ 23. The Excluded Middle, as we saw before, is a peculiar case of the disjunctive judgment; and I think this insight may serve us further to dispel some illusions which have gathered round it.
In the first place we must not think it is a formula, by applying which we can magically conjure elements of know ledge from the unknown deep. It is nonsense to say that it gives us a revelation that any subject must have one of two predicates. For, even if we do not make a logical mistake and really have got contradictory qualities, that is still not the right way to put the matter. Denial is not the predication of a contradictory; and all that Excluded Middle tells us is that, given any possible element of knowledge, you must be right in either affirming or denying any suggestion that is made about that.
We learnt, in our chapter on the Disjunctive Judgment, that this judgment must assume the existence of its subject,18 though that subject may not be the grammatical subject. And when, in the case of Excluded Middle, we are told it will guarantee us the truth of either b or not-& as a predicate of A, we naturally ask, " But what guarantees to us the existence of A? " And we get no answer. Things in themselves either are b or are not b. Undoubtedly so, but what is the real sub ject of this statement? It perhaps after all is not " Things-in- themselves," but is ultimate reality, which may totally reject the whole offered synthesis. In this case we shall at once be able to say that Things-in-themselves are not anything at all in the real world, though, considered as illusions, they no doubt CHAP. V THE PRINCIPLE OF EXCLUDED MIDDLE 155 have qualities. On the other hand, if Things-in-themselves are taken as such to have existence, then that is not proved by our Excluded Middle, but is a sheer assumption on which we base it and which it presupposes.
§ 24. But when we are told, " Between the true and the false there is a third possibility, the Unmeaning"19 (Mill, Logic, II. vii. § 5), we must answer, " Yes, an unmeaning pos sibility, and therefore none at all." The doctrine that proposi tions need neither be true nor yet be false because they may be senseless, would introduce, I agree, " a large qualification " into the doctrine of the Excluded Middle. But I am inclined to think that this " qualification " might be larger than it seems to be, and might be operative perhaps beyond the limits so sparingly assigned to it. But surely, on the one hand, it is clear that a proposition which has no meaning is no proposi tion; and surely again, on the other hand, it is clear that, if it does mean anything, it is either true or else false. And when a predicate is really known not to be " one which can in any intelligible sense be attributed to the subject " — is not that itself ground enough for denial? 20 But logicians who actually (Mill, loc. cit.) are ready to take divisible finitely and divisible infinitely as contradictories, are justified in expecting extra ordinary events. Suppose these terms to be absolutely incom patible, that would hardly bring them under Excluded Middle, unless we are prepared to formulate the axiom thus: When ever predicates are incompatible, then, although there be three or more possibilities, it is certain that one of these two possi bilities must always be true. But perhaps this " qualification " might tend to create more difficulties than it solves.
§ 25. If we turn from these somewhat elementary mis takes, and consider the amount of actual knowledge vouch safed to us by the Excluded Middle, I hardly think we shall be much puffed up. We must remember that, even if we are able to assert about such a subject as Things-in-themselves, we must always be on our guard against an error. We may be affirming about the meaning of a word, or about a mere idea in our heads, and may confuse these facts with another kind of fact (p. 42). But, even supposing we keep quite clear of this mistake, yet when we come to negative judgments THE PRINCIPLES OF LOGIC BOOK I there is ambiguity, unavoidable and ceaseless, about the positive ground of the denial. We may penetrate so far into hidden mysteries as perhaps to be privileged solemnly to avouch that Things-in-themselves are not three-cornered, nor coloured rose-red, nor pock-marked nor dyspeptic. But what does this tell us? What more should we know, if we spent our breath and wasted our days in endless denials of senseless sugges tions? If the ground of negation remains the same,21 each particular denial asserts nothing in particular (Chap. III. pp.
§26.22 Confined to its limits the Excluded Middle is rigidly true. But you may easily assert it in a shape which would exhibit a parallel falsehood to those we considered in examining the Principles of Identity and Contradiction. " Everything," we might say, " is either simply the same as any other, or else has nothing whatever to do with it."
Once again, in conclusion, I must call attention to the positive principle which underlies the Excluded Middle. We assume that every element of knowledge can stand in some relation with every other element. And we may give this, if we please, a metaphysical turn, though in doing so we go beyond the equivalent of the Excluded Middle. We may say, If the real is harmonious and individual, it must exist in its members and must inter-relate them.
§ 27. I may notice by way of appendix to this subject a somewhat subtle argument of Professor Jevons, which I regret to state I am unable to understand. He argues * that to say " A = B or b " must be incorrect. For the negative of " B or b " will be Bb, and by consequence a, the negative of A, must itself be Bb. And the objection to this is that Bb = o. But because " every term has its negative in thought," therefore the negative of A can not be = o, and the premise " A = B or b " is thus indirectly proved false. Professor Jevons proceeds to draw from this a general conclusion that any judgment, in the form " A = B or b" is necessarily erroneous, and that we must write instead of it " A = AB or A&."
Though I fully agree with this last result, yet Professor Jevons' reasoning, as I understand it, appears to me unsound, * Principles, p. 74. For the meaning of Professor Jevons' sym bols I must refer to his work.
CHAP. V THE PRINCIPLE OF EXCLUDED MIDDLE 157