A mere logical negation,15 it is fully admitted by the dia lectical method, need not express a real relation. And, this being so, it seems the better course to consider it by itself as merely subjective, and to express the real implication of exclusives by an affirmative judgment, which sets forth that fact. What denial tells us is merely this, that, when we bring the discrepant up, it is rejected. Whether what repels it is entirely independent, or whether it has itself produced or solicited what it excludes, is quite irrelevant. And it is still more irrelevant to ask the question if the first rejection is merely coquettish, and will lead in the end to a deeper sur render. This all goes beyond what denial expresses, for that, merely by itself, is not asserted beyond our minds.
The dialectical method, in its unmodified form, may be untenable. It has, however, made a serious attempt to deal with the relation of thought to reality. We can hardly say that of those eminent writers who are sure that logic is the counterpart of things, and have never so much as asked them selves the question, if the difference and identity, with which logic operates, are existing relations between actual phenomena.
§ 15. To resume, logical negation always contradicts, but never asserts the existence of the contradictory. To say " A is not B " is merely the same as to deny that " A is B," or to assert that " A is B " is false. And, since it can not go beyond this result, a mere denial of B can never assert that the contradictory Not-B is real. The fact it does assert is the CHAP. Ill THE NEGATIVE JUDGMENT 123 existence of an opposite incompatible quality,* either in the immediate or ultimate subject. This is the reason why the sug gested A — B is contradicted; and it is only because this something else is true, that the statement A — B is rejected as false. But then this positive ground, which is the basis of negation, is not contradictory. It is merely discrepant, oppo site, incompatible. It is only contrary. In logical negation the denial and the fact can never be the same.
§ 1 6. The contradictory idea, if we take it in a merely negative form, must be banished from logic. If Not- A were solely the negation of A, it would be an assertion with out a quality, and would be a denial without anything positive to serve as its ground. A something that is only not some thing else, is a relation that terminates in an impalpable void, a reflection thrown upon empty space. It is a mere nonentity which can not be real. And, if such were the sense of the dialectical method (as it must be confessed its detractors have had much cause to suppose),18 that sense would, strictly speaking, be nonsense. It is impossible for anything to be only Not-A. It is impossible to realize Not-A in thought. It is less than nothing, for nothing itself is not wholly negative. Nothing at least is empty thought, and that means at least my thinking emptily. Nothing means nothing else but failure. And failure is impossible unless something fails; but Not-A would be impersonal failure itself (§ n).
Not-A must be more than a bare negation. It must also be positive. It is a general name for any quality which, when you make it a predicate of A, or joint predicate with A,17 removes A from existence. The contradictory idea is the universal idea of the discrepant or contrary. In this form it must keep its place in logic. It is a general name for any hypothetical discrepant; but we must never for a moment allow ourselves to think of it as the collection of discrepants.
§ 17. Denial or contradiction is not the same thing as the assertion of the contrary; but in the end it can rest on nothing else.18 The contrary however which denial asserts, is never explicit. In " A is not B " the discrepant ground is wholly unspecified. The basis of contradiction may be the assertion A-C or A-D, C and D being contraries of B. But *0n the nature of incompatibility see more, Chap. V.
I24 THE PRINCIPLES OF LOGIC BOOK I again it may perhaps be nothing of the sort. We may reject A-B, not in the least on the ground of A, but because A itself is excluded from reality. The ultimate real may be the subject which has some quality discrepant with A-B. For contradiction rests on an undetermined contrary. It does not tell us what quality of the subject excludes the predicate. It leaves us in doubt if the subject itself is not excluded. Something there is which repels the suggestion; and that is all we know. Sokrates may be not sick because he is well, or because there is now no such thing as Sokrates.
§ 18. Between acceptance and rejection there is no middle- point, and so contradiction is always dual. There is but one Not-B. But contrary opposition is indefinitely plural. The number of qualities that are discrepant or incompatible with A, can not be determined by a general rule. It is possible of course to define a contrary in some sense which will limit the use of the term; but for logical purposes this customary restriction is nothing but lumber. In logic the contrary should be simply the discrepant. Nothing is gained by trying to keep up an effete tradition. If a technical distinction can not be called necessary, it is better to have done with it.
§ 19. Contradiction is thus a " subjective " process, which rests on an unnamed discrepant quality. It can not claim "objective" reality; and since its base is undetermined, it is hopelessly involved in ambiguity. In " A is not B " you know indeed what it is you deny, but you do not say what it is you affirm. It may be a quality in the nature of things which is incompatible with A, or again with B. Or again it may be either a general character of A itself which makes B impos sible, or it may be some particular predicate C. That "a round square is three-cornered," or that " happiness lies in an infinite quantity," may at once be denied. We know a round square, or an infinite number, are not in accordance with the nature of things. But " virtue is quadrangular," or " is mere self-seeking," we deny again because virtue has no existence in space, and has another quality which is opposite to selfishness.
" The King of Utopia died on Tuesday " may be safely contradicted. And yet the denial must remain ambiguous. The ground may be that there is no such place, or it never CHAP. Ill THE NEGATIVE JUDGMENT 125 had a king, or he still is living, or, though he is dead, yet he died on Monday. This doubtful character can never be re moved from the contradiction. It is the rejection of an idea, on account of some side of real fact which is implied but occult.
§ 20. We may conclude this chapter by setting before our selves a useful rule. I think most of us know that one can not affirm without also in effect denying something. In a complex universe the predicate you assert is certain to exclude some other quality, and this you may fairly be taken to deny. But another pitfall, if not so open, yet no less real, I think that some of us are quite unaware of. Our sober thinkers, our discreet Agnostics, our diffident admirers of the phenome nal region — I wonder if ever any of them see how they com promise themselves with that little word " only." How is it that they dream there is something else underneath appear ance, and first suspect that what meets the eye veils some thing hidden? But our survey of negation has taught us the secret, that nothing in the world can ever be denied except on the strength of positive knowledge. I hardly know if I am right in introducing suggestive ideas into simple minds; but yet I must end with the rule I spoke of. We can not deny without also affirming; and it is of the very last importance, whenever we deny, to get as clear an idea as we can of the positive ground our denial rests on.
ADDITIONAL NOTES ADDITIONAL NOTES 1 This chapter contains some serious erors. I have since accepted in the main Dr. Bosanquet's account of negation. See his K & R and Logic. I have briefly discussed the whole matter in T. E. VI.
2 "That subject," i.e., as in one with a selected determination. See s The abstraction of the idea from all " reference " is not defensible. See on Chap. I, § 10. There is always some region in which an idea is real. It is only where the perceived world is taken as the one real object, that other worlds are merely ''subjective" (§13).
As to whether affirmation and denial are co-ordinate, we ^may say that in the end they are so, because the conscious use of ideas as ideas implies both a positive and negative aspect. But denial can be called more "reflective," in the sense that we become aware of it later. We must retain an excluded idea before we can know it 126 THE PRINCIPLES OF LOGIC BOOK I as excluded. The beginning of affirmation, we may say, is an object before me changed ideally so as to lead to action. The beginning of negation is the exclusion of an ideal change in the object — this exclusion not being retained by the mind, though action is thereby prevented. By "action" (I should add) is not meant necessarily action which is "practical." Thus it is not true that we have a separate suggestion and then consciously apply it. The attempt to identify may at first appear to us not as an attempt, but simply as the actual exclusion, where not the actual qualification. It is when we hold the suggestion, while excluding it from our perceived and selected object, that we first have denial in the proper sense of the word.
4 The " suggested synthesis " (here and lower down) needs correc tion in the sense of the foregoing Note.
5 It is true that a excludes b because it is a. It is true that there is a ground and a Why, and that in the end you can not make this Why explicit. But the same holds good also of b, as distinct from and so as negative of a. On the other hand this two-sided negation is at first implicit only and does not appear. You begin positively (as we saw above) with a designated object (Ro) qualified further ideally. It is only later and through reflection that, instead of such an object, Ro (ab), we arrive at a world qualified everywhere by distinctions, at once connected with and opposed to one another, and R so can write our object as / \ 6 1 have, here and everywhere, altered " disparate " where in the original text it was used wrongly for " discrepant." I am quite unable to account for this mistaken use, which, I am sorry to add, recurs frequently, and for the sake of the reader has been now throughout corrected.
7 "Colourless and dark." If "dark" meant "visibly dark"— which I do not think it did mean— there would be a mistake here. See Bosanquet K & R, p. 247.
8 On the subject of Incompatibility the reader is referred to Ap pearance, Appendix, Note A, and to Bosanquet's Logic.
9 These distinctions are (i) exclusion by a specified incompatible; (ii) exclusion of a quality from a space in a subject where that quality is looked for; (iii) exclusion from an assumed space taken as empty on the ground of absence, i.e. of my failure to find the quality there. If you were to drop the assumption made here, and were to reject the empty space, as being either meaningless or itself for some known reason excluded, the above exclusion would become sound. But at the same time it would cease to rest upon failure and mere privation. What on the other hand damns the privative judgment, as ultimate, is its assumption, based on mere ignorance, of an empty space in the character of the Universe. Where however you know positively that the Universe is in a certain respect deter- minable further, there your failure to find a particular qualification CHAP. Ill THE NEGATIVE JUDGMENT 127 (a) is a ground for denial, just so far as you have reason to think your knowledge complete. But see the Notes on Chap. VII, §§ 13 and 28, and see T. E. VII. And cf. Appearance and Essays, the Indexes, s. v. Privation.
10 " Extraneous psychological relation " should be perhaps " a dis tinction turned into a separation and made an exclusion on a mere extraneous psychological ground."
11 " The unexpressed quality." See on Chap. II, § 50.
12 " Fact " here should be " perceived fact." And negation is " sub jective " in the sense that mere negation, mere exclusion, is an ab straction and is by itself really nothing at all. Cf. §§ 15-19. Otherwise negation is not "subjective," though it is more "reflective" than is affirmation (§2).
13 " Nothing is added by additional exclusions." It is true that the abstract negation takes no account of the " how," which therefore, so far, may be the same. But to go beyond this is wrong (Chap. I, 14 " Dialectical view." But, apart from this, in logic we may and must insist that Reality has to be regarded as a disjunctive totality, as the positive unity of diversities each of which is one and is not the others. In our intellectual world we must take every element as within a whole, and as qualified by its relations in that whole, and, further, as qualified by them internally. By "internally" is meant that the element itself, and not merely something else, is qualified. Hence everything will imply its relations both positive and negative. On the other hand we must not say of anything that it is nothing beyond its implications — even though what else it is we are unable in the end to state. The problem of identity and diversity is, I agree, not in the end soluble (see Essays, pp. 240, 264). And our whole world, as merely intellectual, is not ultimately real.
15 " A mere logical negation." The mere must be emphasized.
16 " Much cause " should perhaps be " some cause."
17 "Or joint predicate." In a sense it never is anything but a joint predicate. See Appearance, Appendix, Note A.
18 The main point is this, that denial means exclusion from and by the real. Mere denial, however, rests on abstract exclusion, which, as abstract, is really nothing. Actually the real excludes because the real is qualified incompatibly, and may be so in a variety of senses, the whole of which variety is ignored by the abstract denial. See on § 13.
CHAPTER IV THE DISJUNCTIVE JUDGMENT * § I. The disjunctive judgment may fairly complain that by most logicians it is hardly dealt with. It is often taken as a simple application of the hypothetical, and receives the treatment of a mere appendage. It is wonderful in how many respectable treatises not the smallest attempt is made to understand the meanings of " if " and of " either — or."
The commonest way of regarding disjunction is to take it as a combination of hypotheses. This view in itself is some what superficial, and it is possible even to state it incorrectly.2 " Either A is B or C is D " means, we are told, that if A is not B then C is D, and if C is not D then A is B. But a moment's reflection shows us that here two cases are omitted. Supposing, in the one case, that A is B, and supposing, in the other, that C is D, are we able in these cases to say nothing at all? Our " either — or " can certainly assure us that, if A is B, C-D must be false, and that, if C is D, then A-B is false. We have not exhausted the disjunctive statement, until we have provided for four possibilities, B and not-B, C and not-C.
§2. But however complete may be the cases supposed, disjunctive judgments can not really be reduced to hypothet ical. Their meaning, no doubt, can be given hypothetically; but we must not go on to argue from this that they are hypothetical. The man who illustrated everything else has touched this point too in the Gentlemen of Verona: Speed. But tell me true, will't be a match?
Launce. Ask my dog: if he say, ay, it will; if he say, no, it will; he shake his tail and say nothing, it will. Speed. The conclusion is then that it will.
Launce. Thou shalt never get such a secret from me but by a parable (Act II. Scene v.).
It is indeed by an indirect process, and by making secret a categorical judgment, that hypothetical can express dis junction.
CHAP, IV THE DISJUNCTIVE JUDGMENT I2Q I do not mean that the " either — or " is purely categorical. I mean that to some extent at least it is categorical, and declares a fact without any supposition. In " A is b or c " some part of the statement is quite unconditional. It asserts a fact without any " if " at all. And when pressed with the objection, " But you can not deny that it is reduced to a com bination of supposals," we need not take long to practise an answer. A combination of hypothetical surely does not lie in the hypothetical themselves. It lies in the mind which combines them together, and surveys the field which together they exhaust. It is nonsense to say you are able to " reduce " a statement to elements of a certain character, when these elements, if taken merely by themselves and without a peculiar mode of union, are able to express nothing like the statement.3 The basis of disjunction, the ground and founda tion of your hypotheticals, is categorical.4 § 3. There is, no doubt, some difficulty about the categor ical nature of disjunctive judgments. " A is b or c; " but this mode of speech can not possibly answer to real fact. No real fact can be " either — or." It is both or one, and between the two there is nothing actual. We can hardly mean to say that in fact A is b or c. On the other hand, we are far from expressing simple ignorance. If we merely said " I do not know if A is b, and I do not know if A is c" that would not be equivalent to the original statement. And that we make an assertion can be shown in this way. If the subject of our predicate " either — or " were proved not to exist, our state ment would be false. It is clear not only that the subject has existence, but that it also possesses some further quality.5 The distinction of the apparent and the ultimate subject, which we had to make in our former discussions, must not here be forgotten. " A is either b or c " need not always imply that A is a fact. For example, I may say that "either A exists or does not exist." The subject here is the nature of things, and this either repels the content A or is qualified by it. But still the assertion remains categorical. Throughout the rest of the chapter I shall take A to stand for the real subject, and the reader must remember that in every case the apparent subject may belong to the predicate, and that what 130 THE PRINCIPLES OF LOGIC BOOK I is asserted respecting A may only be true of the ultimate sub ject.
And the same remark applies to such examples as " Either A is B or C is D." The subject in this case is not A or B or again C or D. The subject is the real, which is qualified by the predicate A-B or the predicate C — D.
§ 4. The assertion in " A is b or c " is not that A is b or c. What then do we affirm? We say in the first place that A exists. In the next place we certainly give it some quality.6 What quality do we give it? If it can not be either b or c, can it possibly be something that falls between them? No, for that would be neither. For instance, grey is not white or black, and it excludes both colours. The predicate of A, while neither b nor c, must not be a quality exclusive of either. It must then be a quality common to both, which is not yet either, but is further determinable as one or the other.
§ 5. If we like to call this basis x, then "A is x " is categorically true. We may in some cases have distinguished x and given it a name, but in other cases it is unnamed and implicit. " Man, woman, and child," have a common basis in " human being." In " white or black " the quality " coloured, and coloured so as to exclude other hues," is the attribute asserted. " In England or America," " alive or dead," com mit us to the statements " somewhere not elsewhere " and " organized being." And so, if we call a man " bad or good," we say at least he is a moral agent. There is no exception to the truth of this rule. Even existence and non-existence have so much in common that, in any sense in which we can use them, they imply some kind of contact with my mind. We have seen (Chap. III.) that there is no pure negation. So, in every disjunction and as the ground of it, there must be the assertion of a common quality, the sphere within which the disjunction is affirmed.
§ 6. But x is not any universal whatever which happens to be common to b and c. It is particularized further. It excludes the opposite of each of these qualities, and can not be the nega tive of " b or c" It is affirmed as fully determined not outside the region which is covered by be. But since b and c, as predicates of A, are incompatible, it can not be both of them. The conclusion remains that it must be one. " One single CHAP. IV THE DISJUNCTIVE JUDGMENT 13!
element of the region enclosed by be " is the predicate common to b and c. And this predicate it is which, in disjunction, we categorically assert of A.
So much is fact and no hypothesis; but this by itself would not be the assertion " b or c." The disjunctive judgment is not wholly categoric. Being sure of our basis, the quality xt upon this universal we erect hypotheses. We know that b and c are discrepant. We know that A is particularized within b and c, and therefore as one of b and c. It can not be both, and it must be some one.7 So much is the fact. To complete the disjunction we add the supposal, "If it is not one it must be the other." If A is not b, it must be c; and it must be b, if it is not c. This supposal completes the " either — or." Disjunctive judgment is the union of hypotheticals on a categoric basis.
§ 7. We shall return to consider this process further, but at present we may pause to correct a mistake. It has been doubted if alternatives are always exclusive.8 " A is b or c" it is said, may be taken to admit that A is possibly both. It may either be be or b or c. And, no doubt, in our ordinary disjunctive statements we either leave the meaning to be gathered from the context, or really may not know what it is that we mean. But our slovenly habits of expression and thought are no real evidence against the exclusive character of disjunction. " A is b or c " does strictly exclude " A is both b and c." When a speaker asserts that a given person is a fool or a rogue, he may not mean to deny that he is both. But, having no interest in showing that he is both, being perfectly satisfied provided he is one, either b or c, the speaker has not the possibility be in his mind. Ignoring it as irrelevant, he argues as if it did not exist. And thus he may practically be right in what he says, though formally his statement is down right false; for he has excluded the alternative be.9 And it is not always safe to be slovenly. It may be a matter of vital moment to make our disjunction accurate and complete, and to know if we mean " A is b or c" or " A is be or b or c." About the commonest mistake in metaphysics is the setting up of false alternatives. If we either admit be as a predicate when b and c are discrepant, or exclude be when b and c are compatible, we are liable to come to most 132 THE PRINCIPLES OF LOGIC BOOK I false conclusions. And the very instance we have quoted above should read us a lesson. It is false that the alternative " either rogue or fool " does never exclude the possibility of both. It is a common thing to make this mistake. When we try to guess a man's line of conduct, we first lay it down he is fool or rogue, and then afterwards, arguing that he is certainly a rogue, we conclude that his conduct will be de liberately selfish. But unfortunately the man has been a fool as well, and was not in any way to be relied on. It is often im possible to speak by the card, but still inaccuracy remains inaccuracy. And, if we do not mention the alternative " or both," when held to our words we certainly exclude it.
If we mean to say " A is b or c or again be" the process of the judgment is very simple. A exists and is further determined. It is determined within the region be. A ex cludes all qualities which are incompatible with b and c and again with be. Within be fall b and c and again be, and nothing else falls there. And since these are discrepant, A is but one of them. So far the fact, and then come the hypotheses. If A as determined excludes b and c it must be be; if it excludes c and be it is b; if it excludes b and be it is c. The number of discrepants is of course irrelevant to the nature of the process.
§ 8. But the inaccuracy we have noticed has a natural foundation. We are accustomed to use " or " with an impli cation, and at times we forget whether " or " stands alone or must be taken as so qualified. I will briefly illustrate. If, in drawing up a rule, I lay down that " the number of tickets being limited, each person shall be entitled to a red ticket or a white one," it is at once understood that the alternatives are incompatible. A ticket means here obviously one at most. But, if I say " No one shall be entitled to pass within this enclosure except the possessor of a white or red ticket," I should hardly be taken to exclude the man who was qualified by both. A ticket means here one at least. And it becomes very easy to misunderstand, and to suppose that " or " in each of these cases has a different force.
But in both cases " or " means precisely the same. In the second, as in the first, it is rigidly disjunctive. But in the second of our instances "or" does not stand alone. It is CHAP. IV THE DISJUNCTIVE JUDGMENT j^ qualified by an unexpressed " if not " or " failing that." And this implication makes a vital difference.
§ 9. The alternatives which are offered are not red and white. I am not to be admitted, given white or red. The entitling conditions (so far as they are contemplated) are firstly " white," and then " red, white failing " or " red with out white; " and it can hardly be maintained that these con ditions are compatible. For, if white is there, then red can not make good the failure of white, and the red, that is specified as excluding white, can not by any means admit its presence. What you mean to say is, Suppose white is there, then cadit quastio; but, if white is not there, red will answer the pur pose. And you express your meaning by assigning two alternatives, " white present " on the one hand, and, on the other hand, " red coupled with the absence of white." And this practically provides for every possibility.
The logical objection which may be raised against it is not that its "or " is partly conjunctive, for this, as we have seen, is a pure mistake. The disjunction is faulty not because it is conjunctive, but because it is incomplete. It ignores the possibility of the co-existence of red and white, and in form it might be construed as excluding it. But the reason is obvious. You are never forced to consider separately this individual possibility, since you can always treat it as a simple case of the presence of white. If "white" really means " white with or without red," and " red " means " red on the failure of white," and if the absence of both is fully provided for, then the disjunction is absolutely complete and exhaustive. And these alternatives (i) white with or without red, (ii) red without white, and (iii) failure of both, are absolutely incompatible.
§ 10. And this I think is the answer to an argument brought forward by Professor Jevons (Principles, p. 73). Against the exclusive character of alternatives he urges an indirect argument. If that were so, he objects, the negative of such a term as " malleable-dense-metal " could not be " not-malleable or not-dense or not-metallic." There would be seven distinct alternatives, and this would be absurd.