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 158 THE PRINCIPLES OF LOGIC BOOK I 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 CHAP. V DOUBLE NEGATION 159 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 I6O THE PRINCIPLES OF LOGIC BOOK I the original judgment, we can not do that by refuting our antagonists. We must show ourselves that reality is know- able. 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.
l62 THE PRINCIPLES OF LOGIC BOOK I 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.
I will put the same thing otherwise. With affirmative judgments possibility is the minimum: with negative judgments impossibility is the maximum. Now it is uncertain (we may so interpret Mr. Venn) if the affirmative xy asserts the maximum (actuality) or the minimum (possibility), but it is certain that it unambiguously denies the nega tive. But, if the negative becomes unambiguous because it is arbi trarily fixed at its maximum degree (impossibility), then surely it is clear that we thereby, and ipso facto, are fixing the affirmative at its minimum degree. For so far at least as the affirmative denies and is not ambiguous, it is so because its minimum is enough. And the fallacy is simple. This minimum is not enough unless the negative is fixed at the maximum. Suppose not-xy to mean " xy does not exist," then " xy is possible" ceases to deny this: for, although xy may not exist, it still can be possible. Again if xy meant " xy is actual," then "xy is impossible" (or, again, "if x then no y") is not its contra dictory, and goes a long way beyond its denial. In short, since not-*y means either d-e facto non-existence or else impossibility, it seems absurd to assert that the denial of this is not ambiguous. And if you mean to fix the meaning of the negative arbitrarily, it seems absurd to shrink from doing the same by the positive.
In conclusion, if we suppose that not-;ry is really meant to assert non-existence, that is to deny the actuality of xy, then the error is palpable. You first say you do not know whether xy asserts existence or possibility, and yet you say it denies the non-existence of xy. But possibility, not affirming existence, of course can not deny non-exist ence, and the whole process disappears unless you rapidly shuffle from one term to the other.
This hidden equivocation soon begins to bear fruit in the curious reasoning which immediately follows (p. 143). If I do not misappre hend Mr. Venn, he tries to make a passage from bare possibilities to a positive existential judgment. I confess his metaphysics take away my breath; and I am bound the more to admire his audacity as he somewhat poses as abjuring "transcendentalism," and likes to take things " in a perfectly matter of fact way." But let us see what this way is. We suppose four possibilities, (i) x with y, (ii) x not-y, (iii) CHAP. V DOUBLE NEGATION 163 y not-*, and (iv) not-* not-y. We have first a conditional assertion of xy, and this destroys (ii). We have next a similar assertion of yx, and this destroys (iii). We have therefore, after this second asser tion, but two possibilities, (i) and (iv).
" Before, the positive possibilities were three in number, now they are reduced to two; for it is implied that everything must be either both x and y or neither of the two. Carrying this process one step further, we see that three such " [i.e. conditional] " propositions would be requisite to establish unequivocally the existence of any one of the four classes. If we expunge xy" [i.e. not-* not-;y] "also, we are then reduced at last to an assertion of existence, for we have now declared that xy is all, viz. that within the sphere of our discussion everything is both * and y" (p. 143).
Now, so far as I can see, we may understand this process in two different ways, but on either understanding the argument is vicious. The first way is to take our possibilities as holding within an exhaus tive disjunction. As Mr. Venn says, we know " that everything must be either xy, or * not-y, or y not-*, or not-* not-;y" (142). The disjunction will rest here on a positive existential proposition, and the inference will be quite correct. But the objection is that, on Mr. Venn's theory, we can hardly assume that we have such a dis junction. At least I do not understand why the assertion, Everything is one of four possibilities, should be able to be taken in its positive meaning. We surely are bound, if we wish to be unambiguous, to take it as denying. And if you take it as denying, it does not prove the conclusion. It asserts that what is not one of four possibilities is non existent (or impossible), but it does not say that anything exists. The possibility of everything is all that is asserted, and from this the argument will not take you to more than the sole possibility of xy. If you start with nothing but possibilities, you can not cross from a bare possibility to actual existence simply on the ground that the other possibilities have sunk into nothingness. At least I am sure " tran- scendentalists " especially would be interested in learning Mr. Venn's "matter of fact way" of accomplishing this exploit.
We thus see that the reasoning can not be based on an affirmative existential disjunction. And without this foundation it is thoroughly unsound. Not-* not-y is to be suppressed by a conditional judg ment, and in its dying struggles is to establish xy as "an assertion of existence." I will not ask what the conditional proposition could be. "If anything exists then xy exists" might answer the purpose; but it would not do so unless it were really unconditional, and covertly contained the very assertion that " xy is actual." And this I think is the alternative to which we are brought: we either completely abandon and throw over our doctrine of the superiority of the nega tive, and avowedly start with an affirmation of existence; or else we prove the existence of xy through a double denial which assumes the conclusion in order to extract it.
We may verify the presence of the same ambiguity in the ex- 164 THE PRINCIPLES OF LOGIC BOOK I traordinary assertion that contrary judgments, such as "All x is y" and "No x is y," can be compatible (145). It is not worth while to enter into a discussion of this matter. They are of course compatible if you allow yourself to play on their ambiguity; but how in that case they can be said to be contrary I have no conception. " The interest ing and unexpected application" is to me, I confess, not anything beyond a confused example of a well known doctrine concerning the relations of possibility and existence. But I confess besides that, I have never been much used " to discuss the question in a perfectly matter of fact way."
I need not mention what seem to me other mistakes of much the same kind. And, beside these, there are some statements in connection with the hypothetical judgment with which I do not agree, but for which, I think, my treatment of the subject has provided sufficiently. I am sorry to be forced, both here and again (Chap. VII.), to empha size my difference with Mr. Venn. And by way of compensation I should like, if he will allow me, to offer a suggestion. If Mr. Venn had not such a horror of "metaphysics" and "transcendentalism," if he was a little less resolved to be " matter of fact," and " discuss the question entirely on scientific or logical ground," I fancy he would have come somewhat nearer a solution of the problems it is his merit to have undertaken. At any rate I suspect his idea of science might have been expanded, and some prejudices as to "matter of fact" have been somewhat loosened. He would certainly have imbibed a dis like for artifices, and such a scruple against entertaining commodious fictions, as in itself would have saved him from a succession of serious logical mistakes.
ADDITIONAL NOTES 1 On the idea of a term being related to itself see Essays, Index, s. v. Terms.
2 "Within a whole of fact." "Fact" is of course to be under stood here in the widest sense.
aAll truth must abstract, and, so far as it is truth, it can not be made false from the outside. How far any truth which abstracts can be wholly true, I have discussed elsewhere. See Appearance and Essaysf the Indexes.
4 "In the judgment S — P." Add "of which we were speaking." And, after "becomes an identity," add "and so enters as an element into a fresh S — P." In the next sentence the "it" (in "belongs to it") is to be emphasized.
5 What the reader should keep in mind is the following. Differ ences are all incompatible if you attempt simply to identify them. They are again all compatible if and so far as they are merely con joined. Wherever there is conjunction there is something more in the conjoined whole than mere identity, so that here the whole, as CHAP. V DOUBLE NEGATION simply identical, does not attempt to enter into each diversity. The whole, however, if it is to be made intelligible, must become dis junctive. The aim of disjunction (see Chap. IV, § i) is to replace the conjunctive unity by the discovery and statement of conditions. As to why certain conjunctions are possible in fact, while others are not so — logic does not enquire. The question of detail belongs here mainly, I think, to psychology. On the above see Appearance, Appen dix, Note A, and Bosanquet's Logic, II, Chap. VII.
6 "The discrepant with A...attribute." This sentence should run, "The incompatible with A is what is not a mere joint predicate with A in any subject, nor is joined to it...attribute."
7 " For denial...affirmation." In this sentence " the " should 8 " From a certain theory." " From " is here, I think, rightly used in contradistinction to " by."
9 The main point here is as follows, Incompatibles exist, and no one denies this fact. And, so far as they exist, the Law of Con tradiction holds. The real question is as to the limits within which, and the conditions under which, incompatibles are found and can be justified. How far in other words is the truth of contradiction, as such, only relative and more or less of an appearance? What, as I understand it, the Dialectical Method is concerned to deny is merely the absolute, utter and final, truth of fixed incompatibles. On the whole matter see my Appearance, Index, s. v. Contradiction.
10 " What I mean, &c." The point here is that, where you have differences in A, A is never mere and bare A. Cf. on § 10.
11 " Stationary contraries "..." are found." Yes, but as an appearance only. See Note 9.
12 On the principle of Excluded Middle, while once more referring the reader to Bosanquet's Logic, I will add a few words. This prin ciple presupposes a disjoined world of incompatibles, and its truth is but relative and limited to Reality taken in the character of such a world. So far as the real is otherwise, as being either below or, again, above the level of disjunction, the principle does not hold. If we accept the view that no truth is quite true and no error merely false — a view advocated in my Essays and Appearance — we must admit that Excluded Middle, however necessary and important, is not true absolutely.
In rejecting it as the principle of disjunction, I meant to deny that disjunction stands upon it in the shape of a ready-made base. We may on the other hand take it as containing the abstract form of disjunction. It is disjunction made all-embracing and dual by group ing all the incompatibles, save only one, under their negative aspect, with the result that nothing is left beyond assertion or denial. The leaving the other members of the whole thus artificially blank, is of course a grave shortcoming. For, merely in the shape of such an abstraction, these other members are not real positively, and so are not real at all. Knowledge is not advanced by the exhaustiveness 1 66 THE PRINCIPLES OF LOGIC BOOK I of disjunction effected formally through an artificial duality. Its real object is to discover in concrete detail the full connection of its elements.
Excluded Middle is, however, in a sense more fundamental, and goes, we may say, further than mere disjunction. For it asserts the actual being of the disjunctive world. We affirm in it that Reality is a region where "either or" holds, and that everything is so de termined as to fall within this sphere — everything, that is, so far as it is not self -contradictory or otherwise senseless. (For the connec tion between these two ideas see T. E. VIII.) But, as was remarked above, we have here a relative truth which is taken wrongly if made absolute.
I may add that the principle that every idea is attributed to Reality, and is therefore in some sense real, has no special connection with Excluded Middle. And the same thing holds again of the corollary that, where all possibles but one are excluded, the one left is actually real.
13 " True or false." See, however, the preceding Note.
14 "On this basis." But see on Chap. IV, §6.
15 " My impotence." See on Chap. Ill, § 9.
16 "Must wait"...fact." But it is better, I think, to take Excluded Middle as assuming, not only connection everywhere through out the Universe, but also that special kind of connection which holds between incompatibles. See Note 12.
17 " False alternative." But, if we say this, surely we must mean that Excluded Middle has been assumed to hold outside its own lim ited sphere, and that hence it does not hold everywhere. Again, in the next paragraph, " fallacious " can not, I think, stand. But I agree that it is certainly possible, and sometimes easy, to object wrongly to the legitimate and necessary use of Excluded Middle.
18 "The existence of its subject." But see on Chap. IV, §3.