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 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 socalled " 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 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 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 notcontinuous.
The ground of the objection to the Excluded Middle is, I am bold enough to think, fallacious. Given not fixed discrepants 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.
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-inthemselves," 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 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 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.
and I can not reconcile his conclusion with his process. I will take the latter point first. It appears to be right to judge " A = AB or Ab" But what is the negative? I suppose the negative is AbE, and we must conclude that a = AbE. But the term AbE most clearly = o. So that, after all, we are left with a conclusion which proves the falsity of our premise.
The result is thus out of harmony with the argument, but for all that the result is perfectly true. It is true that we can not say " A = B or b" and I will proceed to show why this must be true. We must take it that A has a determinate quality; but what is merely B or & is anything whatever. Eb being nothing, what is simply not-Eb will therefore be any thing. And, as A is something definite, " A = anything " will of course be false. The sphere " B or b " is wholly un limited.
This confirms the doctrine we have above adopted (p. 123). If you take not-B as the bare and simple negation of B, it is nothing at all. And if you keep to this sense, then " A =• not-B " could not be true. The true meaning of not-B is any indefinite general quality which does exclude B. And, so long as A is something definite, A can not be this. I am inclined to think from the presence of x (Principles, pp. 94, 95) that Professor Jevons would agree with this doctrine.
But the conclusion, which Professor Jevons uses as false, is not only quite true, but is the necessary result of the true doctrine he accepts. Taking A as the genuine subject 23 that lies at the base of the disjunction, then " a = nothing " must follow at once, since " A is B or not-B " does assume and postulate that A is real. If a were anything but non-existent, you could not use A as the base of a disjunction. What is wrong is not this conclusion or its premises, but the mistaken idea about the negative which Professor Jevons has em braced.
I confess I am not sure if I apprehend him rightly, but he seems to argue that the non-existent is not thinkable, and hence, because the negative of everything is thinkable, you must never have a negative which is non-existent. Now I admit that, if " existence " is used in the widest possible sense, this argument is tenable. The unreal, the impossible, and the non-existent will every one of them exist, provided they are thinkable. And, since even nothing itself 24 in this sense exists, it is obvious the whole argument thus disappears.
But, if it does not disappear, and if existence be taken in anything like the sense of reality, the argument becomes vicious. We have no right to assume that the contradictory of an idea which is true, must itself be real. Take for in stance the idea of " reality " itself. I could not even admit that in thought all ideas are qualified by their negations. I should doubt if the highest term we arrive at can be said to have an opposite even in thought, although by an error we are given to think so. But to hold that what contradicts the real must be real, is a logical mistake which I cannot venture to attribute to Prof. Jevons.
I may end with the remark that it would be entertaining and an irony of fate, if the school of " Experience " fell into the cardinal mistake of Hegel. Prof. Bain's " Law of Rela tivity," approved by J. S. Mill, has at least shown a tendency to drift in that direction. " Our cognition, as it stands, is explained as a mutual negation of the two properties. Each has a positive existence because of the presence of the other as its negative" (Emotions, p. 571). I do not suggest that Prof. Bain in this ominous utterance really means what he says, but he means quite enough to be on the edge of a preci pice. If the school of " Experience " had any knowledge of the facts, they would know that the sin of Hegel consists, not at all in the defect, but in the excess of " Relativity." Once say with Prof. Bain that " we know only relations "; once mean (what he says) that those relations hold between posi tives and negatives, and you have accepted the main prin ciple of orthodox Hegelianism.
§ 28. It is obvious that duplex negatio affirmat. To say " It is false that A is not B " is equivalent to the positive assertion, " A is B." But this is not because the added negation barely negates the original judgment. For if that were all, we should be left with nothing. If mere not-A is simply zero, then not-not-A is, if possible, less. And we must not say that negation presupposes a positive judgment, which is left in pos session when the negative is negated. For we saw before (Chap. III. §4) that this positive judgment is not presup posed.
§ 29. The real reason why denial of denial is affirmation, is merely this. In all denial we must have the assertion of a positive ground; and the positive ground of the second denial can be nothing but the predicate denied by the first. I can not say " It is false that A is not b" unless I already possess the positive knowledge that A is b.2Q And the reason of my incapacity is that no other knowledge is a sufficient ground.
§ 30. I will briefly explain. We know well by this time that, in judging A not to be b, I presuppose a quality in A which is exclusive of b. Let us call this y. I now desire to deny my judgment, and need, as before, some quality as the ground of my new denial. Let us take some quality other than b. Let this quality s be exclusive of y, and let us see what we have. We have now A^ with the exclusion of y which excluded b. But that leaves us nowhere. We can not tell now if A is b, or is not b, because z itself, for anything we know, may also exclude bf just as much as y did. What, in short, we have got is our own private impotence to deny " A is b "; but what we want is an objective ground for declaring such a denial to be false.
The same result holds good with any other quality we can take, excepting b itself. The only certainty that b is not absent is got by showing that b is present. For the possible grounds of the exclusion of b being quite indefinite, you can not get rid of them by trying to exhaust the negations of b. You could only do that if the number of possibilities with respect to A had already been limited by a disjunctive judg ment. And this is not here the case.
Suppose, for instance, we have the judgment that " Ulti mate reality is not knowable," and we wish to assert that this judgment is false. We expose the ground on which it is based, and go on to show that this ground is not valid. Our pro ceeding, no doubt, may be perfectly admirable, but all that it gives us is the right to doubt the original judgment, and to deny the truth of the basis it stands on. If we wish to deny the original judgment, we can not do that by refuting our antagonists. We must show ourselves that reality is knowable. The ground for the denial of " A is not b" must lie in § 31. I will endeavour to remove a possible source of mis apprehension. It might be urged that in practice the denial of a judgment can always be denied by something other than the judgment itself. Thus, for instance, " It did rain yes terday," may be false, because it snowed or because it was fine. But each of these can be denied on the ground of the other. The result of our double negation of " it rained," might be either " it snowed," or again " it was fine ": and we might return to " it rained," by virtue not of a double but of a triple denial.
But this objection would rest on a misunderstanding. It is perfectly true that, in denying " it rained," I must imply and make use of some discrepant quality. It is, once more, true that what I have in my mind, and should assign as my reason, may be either " it snowed " or again " it was fine." But it is a mistake to conclude that the denial really rests upon either the one of these or the other. Whatever you might have had in your mind, no logic could force you to allow that your denial had committed you to either " it snowed " or " it was fine." What we use in denial is not the whole discrepant: it is that part of the discrepant which answers our purpose. The denial asserts no more than the existence of so much quality as is enough to exclude the judgment " it rained."
This universal " so much " is possessed by either " it snowed " or " it was fine," and this you can not banish by anything short of the judgment " it rained." In other words, if you say " it did not rain," you are at once committed to a positive " because," but you are committed to nothing but an unspecified quality. The evidence for this quality no doubt in the end must be found in the presence of a contrary asser tion, but the mere contradiction does not affirm this or any particular contrary. It affirms merely some contrary, and you get rid of this only by the judgment " it did rain." We find here once more the constant ambiguity, which we have seen (Chap. III. § 19) makes the use of negation so precarious. It is so difficult to work with double denial that I hardly can CHAP. V DOUBLE NEGATION l6l expect in the present volume to have supplied no example of the error I condemn.* * Mr. Venn, I think, has certainly done so.28 When I had the pleasure of reading his Symbolic Logic, I congratulated myself on the fact that I had already written the present and all the preceding chapters. I have not found occasion in consequence to alter anything of what I had written, but I should like to use one of his principal doctrines to exemplify the fallacious use of the negative. I have added this discussion as a mere appendix, for it hardly carries the subject further. It is due to myself to defend my own views against a counter theory from a writer of established and merited reputation.
After calling attention to the ambiguity of affirmative universals, the doubt, that is, if they affirm the existence of their grammatical subject, Mr. Venn, if I understand him rightly, asserts that at all events the negative is not ambiguous (p. 141). I will not here enquire if in other places he is compelled to recognize that the opposite of this assumption is true. At all events the foundation he here seems to build on is the assertion that negatives have only one meaning. " It comes to this therefore that in respect of what such a proposition affirms it can only be regarded as conditional, but that in respect of what it denies it may be regarded as absolute" (142). The affirmation of xy is always ambiguous, since x may not be actual; but the denial of x not-y is perfectly clear. And upon this basis he seems to build his doctrine.
Now the reader of this volume will know that a negation is always ambiguous. We may consider this as settled, and I will not re-discuss the general question. I will first call attention to the seeming absurdity of Mr. Venn's doctrine. He teaches in effect that, although you do not know what a statement means, you can always tell what you mean by denying it. And he ought to hold that the ambiguity of a judgment at once disappears, if you deny it and then deny your denial. This course has not generally been found so successful.
But it is better to show the actual mistake. And we will preface our criticism by setting down some elementary truths. You can not argue from the assertion of possibility to the assertion of actuality, but you can always argue from the denial of possibility to the denial of actual ity. To deny possible x (you must of course not take "possible" as "merely possible") is by implication to deny actual x. Now the simple application of this commonplace doctrine is that, if you are given a connection xy and do not know whether it is possible or actual, at all events, if you deny its possibility, you may be very sure that you also, and as well, have denied its actuality. This is literally (unless I mis understand him) the whole principle which Mr. Venn unconsciously proceeds upon, and the idea that it could lead to any great result, or to a better understanding of hypothetical, seems somewhat strange.^ I can not be quite sure of his exact procedure, but I think it is this.
The affirmative judgment both affirms and denies. Mr. Venn will not say that what it affirms is mere possibility, but he quietly assumes that what it denies is impossibility. (If he does not do this, he makes a simpler mistake to which I will return.) That is to say, he tacitly and without any justification assumes that x not-;y asserts the impossibility of xy; and it is solely by denying this arbitrary fixture that the positive xy becomes unambiguous. But if he wishes to restrict the affirmative judgment to the minimum sufficient to deny the denial of possibility, surely it would be better to say at once, "The affirmative judgment does not assert more than bare possibility." He would so have done openly and in an intelligible manner the very thing he has in effect done, indirectly and most objectionably, by going round through two denials. The procedure could in no case have become more arbitrary.