i There is a main point in this Chapter where, if not correction, at least some explanation seems necessary. All inference depends on the unbroken individuality of a single subject; and in this sense all inference may be said to fall under the category of subject and attribute (Bk. II. I. VI. §13). But, so understood, this category must not be taken as merely one among others. It is pre-supposed through out as the condition of the rest, which, as against it, will be subordinate and special. On the other hand there are inferences which are made simply under and by the use of the above category in an individual case. This will be true, for instance, of the entire syllogistic logic. Hence to say that on my view logic is confined to the sphere of sub ject and attribute, or substantive and adjective, would be true or false 272 THE PRINCIPLES OF LOGIC BOOK II. PT. I according to the sense given to such a statement. Everything, I agree, must with me fall under this main principle, and I know of no other main principle which to myself is intelligible. But to add that on my view the other special categories are not necessary, and that the conclusions got under these categories could, so far as correct, be got without them, would, I submit, be untrue. And I am bound to claim whatever merit is due to me for having insisted on the opposite. But this double sense in which, in this Chapter and elsewhere, the category of subject and attribute may be said at once to preside and yet to be co-ordinate, is, I admit, misleading. And this double sense, if borne in mind throughout these pages, tends, I think, to make part of the detail superfluous.
The reader should further keep in view the following distinctions. We have, first, the knowledge that everything falls within and qualifies one individual Universe. We next, in any particular case, have to do also with some subordinate individual whole. Now, so far as this whole is not taken as known immediately, all the elements within it must be somehow interrelated. They all are at least related among themselves as common adjectives. Further, having distin guished the adjectives within the whole, you can go on to qualify this whole by what beyond it is true of any of these adjectives — so long, that is, as you do not, by a further abstraction, set free and substantiate this adjective. And so again, subject to the same con dition, you can infer similarity (§3).
But the knowledge you so far possess does not enable you to draw conclusions under the more special categories — such as Space or Time or Degree. These have powers and rights of their own, though, while acting by and under these, you can, at the same time and concurrently, make use in addition of your power under the more general category of subject and attribute. But, so long as we remain clear in principle, the effort to distinguish in detail the pre cise limits as to where, in this or that case, the above concurrent use comes in, seems really superfluous.
2 " So far as is required &c." Cf. Bk. II. I. II. § 9.
3 " Is not the actual &c." " Is not by itself the actual &c.," would have been better. And (lower down) "anything which we shall have" is better, I think, than "anything which we have." Again lower down, for "get a definite relation" substitute (in the first sentence) "a more definite relation," and (in the second sentence) substitute "a new direct relation." And (still lower down) after "not that general connection" add "nor even anything that follows from the mere co-inlierence of A, B, and C in a new apprehended whole."
4 The statement "in the first of these &c.," seems clearly wrong. And, otherwise, the conclusion would become illegitimate.
5 " The categories...footing." See § i. And cf. the Index.
6 To " We should then have become circular " add " or should have failed altogether." So far, that is, as you take this attribute and CHAP. IV THE PRINCIPLES OF REASONING 273 this subject as immediately one, you remain within one individual whole of immediate qualification.
7 Here Nos. IV and V must be corrected by the insertion of " within one world of time or space " after " Where one and the same term." We can not assume the spatial or temporal unity of all spaces or times. (See Appearance, the Index.) Whether a similar correction should be made in No. Ill will depend on the sense which there is given to " by virtue of one and the same point."
The real question everywhere is whether the consequence is the self -development of that which we take as the subject, or whether, by the intrusion of something foreign, the identity of the subject is broken. When we are asked by a writer in Mind (I regret to have lost the reference), whether in "A cheats B, B cheats C, and therefore A cheats C," we have a valid inference — the answer is easy. We have here a good inference if we take the action of A on B as itself developing itself, without loss of identity, through B into action on C. On the other hand the inference is vitiated, so far as we suppose a foreign condition to be necessary, such as to destroy the process when viewed as the self-development of A's action.
In connection with the Synthesis of Degree I may remark that, though I failed in this volume to notice what is called the argument a fortiori, I should at once have placed it under the above head. The argument obviously depends on the comparative amount of ground.
s "Private judgment" should be "individual judgment"; and a similar correction should be made in the last words of this section.
9 " For you do not know, etc." The " for " seems here to involve some confusion. When (see Note 7) the axiom has been corrected, it would be better to substitute for " the two space-relations belong to one world " the words " the premises fall under the axiom."
10 It would, I think, have been better if this attack upon Casuistry and Hedonism had been shortened, if not omitted.
CHAPTER V NEGATIVE REASONING1 § i. The general nature of negative reasoning does not vitally differ from that of positive. We have, given us in the premises, two or more relations presenting us with certain identical points, and on the basis of these points we combine the relations into an individual whole. We then by inspection find a new relation within that whole. The conclusion rnay connect two terms directly, as in A — B — C. *. A — C, or it may connect them indirectly, as A — B — C. •. A — (B)C, or A(B) — C.2 The new line that is drawn may fall clear of the middle-point of the construction, or may pass through it on the line of the old relations. Negative reasoning and positive have all these qualities in common. It is true that in a negative inference the line that connects the terms of one relation is a line of denial; one part of the figure, which ideally we con struct, consists of a repulsion; and the fresh connection we draw from that construction is a connection by exclusion. But these differences are varieties within the same main principle.
§ 2. It might seem as if nothing remained for us to do but to state and illustrate those negative formulae which corre spond to the axioms of affirmative reasoning. And to this we shall at once proceed to address ourselves; but it is right to premise that there are further difficulties which lie in wait for us at the end of this section.
In negative reasoning we may so state the principle,3 " If B is related within one genus positively to A and negatively to C, then A and C are negatively related within that genus. And if the affirmative and negative relations (A — B, B — C) are heterogeneous, yet, if one is in the category of subject and attribute, there is a negative inference within one or both of the two categories which have appeared in the premises." Unless A — B B — C are within the same genus, or unless one is a relation of subject and attribute, there is no connection at all.
CHAP. V NEGATIVE REASONING 275 I. Synthesis of subject and attribute.
(a) Where the attribute is not taken as distinct from every subject, what is denied of the attribute is denied of the subject, and where the attribute is denied the subject is denied.
(b) Where the subject is not taken as distinct from every attribute, what is denied of the subject is denied of its attri butes, and where the subject is denied then, in that sense, the attribute is denied.
(c) Where two subjects have the same or a different attri bute, they are so far not different or not the same.
Examples: (a) "A triangle has not got two right angles; this is a triangle, and has therefore not two right angles." " A rectangular triangle is not equilateral; this figure is equilateral, and therefore can not be a rectangular triangle." (b) " Man is not a quadruped, man is a mammal, therefore a mammal may be (the human mammal is) not a quadruped; and a quadruped is not a mammal in every sense of that adjective." (c) " My horse is vertebrate, this animal is a worm, and there fore is not the same as my horse."
II. The Synthesis of Identity must become a Synthesis of Identity and Difference, "Where two terms have the same point in common, and one of them by virtue of this point is different from a third, there the other and the third differ in this same point."
Example: "A piano (A) is in tune with B, which is not in tune with C, and therefore A and C are not in tune with each other."
In the Synthesis of Degree, of Space, and of Time, we have no occasion to alter the formulae. We may give as examples, III. A is as heavy as B, B is not lighter than C, therefore A is not lighter than C.
IV. A is not before B in time, B is contemporary with C, therefore A is not before C.
V. A is due east of B, C is not north of B, therefore C is not north of A.
§ 3. We seem to have performed our task successfully, but must deal with a further complication. We may be taken to have sinned against two prominent rules of the traditional logic, since on the principles we have given you may get a conclusion from two negative premises, and that conclusion 276 THE PRINCIPLES OF LOGIC BOOK II. Pi. I may at least in part be affirmative. Yet I can not reject these traditional rules as errors, and if they have committed over sights is a question which turns on their interpretation. With out doubt if you interpret negative premises strictly, that is, take them in the shape of bare denials, then the rule which forbids an inference is valid. And the second rule, which confines the conclusion to a mere denial, is without doubt valid unless you break through another syllogistic precept. If you insist on eliding the middle term, then not only must the result be partly negative, but it really is limited to a judgment which denies. And thus, if in their statement the rules turn out to have gone too far, they at all events have been based on a solid foundation.
It is not hard to understand this; from two bare denials there can come no conclusion, because there can not be any construction. Why no construction? Because there is either no common point, or, if there is a common point, because you do not know the position of the other terms. Let us take the last first; in negative reasoning we may represent the denials by lines of exclusion; but, if we interpret the premises strictly, we find ourselves unable to give these lines any definite posi tion. A is not C nor B, but the exclusion of C and the ex clusion of B, though we represent them truly by lines of rejec tion, fall we know not where. The excluded has got no determinate position, and therefore no known relation to other elements.
And this is not all, for if we wish to see the real state of the case, we must go back to our doctrine of the negative judgment (Bk. I. Chap. III.). A mere denial does not in any way give existence or position to the thing it denies.4 Thus in " A is not B " we assert the simple rejection of B by an unstated quality belonging to A, and in respect of B we know nothing at all but its banishment from our universe. But it is obvious that, when a term is so banished, we know about it nothing definite save its rejection by A. No matter then how many negative premises we may have, since by adding to the number of our banished terms we do not get any nearer a conclusion. The exiles do not move in any real world at all, and to unite them by a line of connection is impossible.
Thus even if two denials have a common subject, we can CHAP. V NEGATIVE REASONING 277 not go from those denials to a further relation.* And we are stopped elsewhere by another obstacle, for we have not got the common centre required for a construction. In " A is not B and B is not C," we have in the one case the exclusion of B, and in the other case the exclusion by B; we have first absence and then presence. And again, if we give our premises another form and say " B is not A and B is not C," we can not go to a relation between A and C, since (apart from other reasons) the quality of B may be quite different in each denial. Perhaps from " C is not A and B is not A " we might be tempted to argue to a positive relation of partial identity between C and B. But here again our centre would be wanting, for we do not know if the quality which ensures the rejection is not wholly different in each of these cases. And thus our premises may furnish a ground for suspicion, but they no more give us proof than would such positive premises as " A is like B and C," or " A is like B, and B is like C." In short given two denials there is either no common point, or else the two relations which start from that centre terminate in nothing which can be related.
The rule which forbids all the premises to deny is thus shown to have a solid foundation; and we may say the same of the rule which prohibits a positive conclusion. For since the predicate denied is completely expelled from the world of the subject, we are left with no relation beside the repulsion. It is clear then that you can not have a positive connection either between the predicate and that which exists in friend ship with the subject, or between the subject and what shares the fortunes of the predicate. In "A — B B — C," if one relation is negative, we can not in any way draw a line A — C which falls outside B. For A and C will be separated in two different worlds, and if one is in any way to come in contact with the other, the line of connection must pass through B. But on one side of B is a mere rejection, and it is therefore evident that a positive line can not be drawn beyond the centre, and that the new relation must add to the rejection which already exists in B. It is indeed not true that this * In "A is not B and not C, therefore B and C are so far alike " the premises are positive. B and C are both discrepant in quality with A, or have the psychical fact of rejection in common.
278 THE PRINCIPLES OF LOGIC BOOK II. PT. I extension is a mere denial, and again it is not true that the conclusion must be wholly negative; but for all that the second traditional rule has, like the first, a rational foundation.
§ 4. But though both the precepts stand on a solid basis, the meaning of the first calls for some restriction, and the second is not true without an exception. Two denials should not give a conclusion at all, and yet you can not say that of two premises which deny. In his Principles of Science, p. 63, Prof. Jevons has called attention to the subject; "Whatever is not metallic is not capable of powerful magnetic influence, (i) Carbon is not metallic, (2) Therefore, carbon is not capable of powerful magnetic influ This argument no doubt has quaternio terminorum and is vicious technically, but the fact remains that from two denials you somehow have proved a further denial. " A is not B, what is not B is not C, therefore A is not C." The premises are surely negative to start with, and it appears pedantic either to urge on one side that " A is not-B " is simply positive, or on the other that B and not-B afford no junction. If from negative premises I can get my conclusion, it seems idle to object that I have first transformed one premise; for that objection does not show that the premises are not negative, and it does not show that I have failed to get my conclusion. And if we leave the limits of the syllogistic logic examples come to us from every side; " A degree A can not be less than B, B is not less than C, therefore C can not be greater than A, or A must be equal to or greater than C; " " Event A is not before B, C is not after B, therefore A is not before C, or C is simultaneous with A or before it; " " C is not north of B, B is not north of A, therefore A is not south of C, or A is due east, or west, or on the north side of C." It is bootless here to fall doggedly back on the technical rules of mood and figure, since, if we keep to these, we can not even prove the positive conclusions from the positive premises. If " A to right of B " is a positive relation of A to B which can not be reduced to predicate and copula, why should we not have in " A not to right of B " a negative relation which is CHAP. V NEGATIVE REASONING 27Q also irreducible? The traditional logic may object to the latter, but it has put itself out of court by first objecting to the former; and, if it is quite wrong in one case, it may be quite wrong in another.
§ 5. In this case it is not wrong, for it happens to be right. The restricted portion of the field it occupies happens here to be the limit of the subject. For denial as such can not fall outside the single category in which the syllogism is shut up.
A denial as such, we have seen long ago, is merely the exclusion of an ideal suggestion, and hence no negative rela tion between positive existences can ever be expressed by a mere denial. But then on the other hand a bare denial can never be found, for, when A excludes some relation to B which is offered in idea, there must always be a ground for that rejection. The base of the rejection must be a positive quality, unspecified but necessary; and hence, wherever we have negative judgment, we have in addition some positive assertion, which may not be explicit but which must be there. And this, as we saw, is such a fount of ambiguity that in denials we seldom know all we are saying (p. 125).
We may verify this in the examples we have used. In the first we assume that A has degree, and upon that basis of positive assertion we proceed, by exclusion of the alternatives denied, to a positive result. In the second the argument really starts from " A is an event with a position in the series after or simultaneous with B." In the third we assume that A falls in space and in a relation to B marked out by exclusion. In all these if we kept to mere denial we could not prove anything, since we may deny " less than B," or " prior to B," or " north of B," of what has no degree and no time and no position. Such a course might be unusual but is legitimate and recognized, because the denial as such covers all pos sibilities.
§ 6. If we take as our rule that from negative premises you can not argue, then, stated so, that rule is incorrect^; and it is false even to say that denials give no inference, since every denial has a positive side. That positive side is latent and may escape us; in " 7 is not less than 5 '+' i, 5 +' i « not less than 4, and therefore 7 is not less than 4," we do not say that 7 is a number at all and must stand in some numerical 280 THE PRINCIPLES OF LOGIC BOOK II. Px. I relation with 5 '+ i. And thus in assuming it we have passed beyond the denial, though not beyond what the denial im plies. It is necessary therefore in expressing our rule to make a distinction. You can not argue, we must say, from two denials, so long as you keep to bare denial. If you treat the assertion which those denials imply, then you are not keeping to the side of denial. And, if we formulate it so, the rule will hold good.
Denial implies removal or exclusion, and from exclusions or removals you can get a conclusion. " Removal of A is removal of B, removal of B is removal of C," gives " Removal of A is removal of C; " and " Absence of A is absence of B, absence of B is absence of C," proves that absence of A is absence of C. But here our real premises are " What re moves A removes B," and " That which is without A is also without B." You can hardly say that these premises are quite positive, but they contain much more than a bare denial. Thus negation must always remain ambiguous (Book I. Chap. III.), for "No A is B," may merely banish B, while again it may assert " The absence of A is the presence of B." " If A is there then B will not be there," and " Since A is not there B must be there " are both expressed by this doubtful formula. But if we confine negation to mere denial it is the exclusion of an idea by an unspecified quality, and if we con fine the denial to its negative side it is the mere exclusion of a suggested idea. It is upon this last understanding that the traditional rule is actually valid.
It would not be valid if negation were assertion. If in " A is not B " the exclusion of B were a condition necessary to the existence of A, then B must be banished if A is to be there, and if B is not there B can not be banished. And from negative premises, if so interpreted, it no doubt might be possible to get some conclusion. But this interpretation we long ago saw was erroneous. The denial excludes an ideal suggestion, and the fact which lies at the base of the exclusion need be no relation of A to B, but on the other hand a quality of A or again of some more ultimate reality. But this quality is latent and wholly unspecified.
§ 7. We have seen that, upon a strict interpretation of negative premises, the first of the rules we mentioned is valid.
CHAP. V NEGATIVE REASONING 28l What then is to become of our principles of synthesis, since they collide with the rule and can not be true? But I think it is better to leave them standing, for they are valid if the sense of negative premises is not confined to what they deny.
Otherwise of course they must be corrected. It is im possible to have any negative inference which will fall wholly within the categories of identity, or time, or space, or again degree. One premise at least must confine itself to the rela tion of subject and attribute.
This is very obvious. One premise must deny, and no denial as such can be referred to any category beyond the relation of attribute to subject. The denial is the exclusion of an ideal suggestion, and a relation of time, or space, or degree falls within this suggestion which the subject repels. It is clear then that the denial of a connection, say of space, is not a connection in the category of space. The subject excludes, it is true, by a quality, but you do not know what that quality is. And since you do not know what quality repels, the repulsion and the quality which forms its basis can not pass beyond the sphere of simple attribution. Thus " A is not north of B," if restricted to denial, means " A repels the suggestion A to north of B; " and we can not possibly take this as anything more than an adjective of A.
If we refer to the examples we gave in illustration (§2), we must so interpret the negative premises. "B is not in tune with C " means " B excludes the attribute of being in tune with C," and " B is not lighter than C " means " B ex cludes a certain relation of degree to C." But of course B might repel these relations with C although it possessed no note at all, and although it had no degree of any kind; and in the same way the denial that B is in such a position may be true though B has no place whatever. If one of the premises be confined to denial that premise is shut up within the category of subject and attribute. m f But having so restricted the character of our premises it is natural to expect a restricted result. Our rule will now be, " In all negative inferences the conclusion is confined within the relation of subject and attribute, unless that conclusio can in any way be affirmative." m §8. But can the conclusion be anything but negative!
282 THE PRINCIPLES OF LOGIC BOOK II. Pi. I This is the question we have next to discuss. The rule for bade an arhrmative result, and we saw that this rule was based upon truth. For since in A — B B — C one relation is negative, A — C can not be joined by a line of connection which passes anywhere except through B. And, since part of this line must consist of an exclusion, we saw that A — C must have a negative character (§3).
The result is unshaken, but it omits a possibility. The conclusion need not take the form of A — C, since the result which we get from the union of our premises, may be found in the whole ideal construction. The syllogistic practice is to elide the middle; but if we do not choose to perform this elision, who on the one hand can order us to do so? And on the other hand who can deny that the result which we obtain is a real inference? " A takes precedence of (is lighter than, sits on the right of) B, B is not younger than C, therefore A takes precedence of (is lighter than, sits on the right of) a person (B) not younger than C." There is here no direct conclusion A — C, and there is again no inference within one category, and at the same time one premise seems to be used as mere denial. On the other hand I see no reasonable ground on which we can deny that we have got a conclusion. Yet this conclusion is neither a mere denial, nor does it fall within the category of subject and attribute.
We may go beyond this. In the syllogism itself, if we decline to elide the middle term B, we may have an inference the conclusion of which is more than a denial. Take an instance in Celarent, "A lung-breathing animal (B) is not a fish (C). All Cetacea (A) breathe by means of lungs (B)." From this the regular conclusion is " A is not C." But " All Cetacea have a quality, viz., breathing through lungs, which excludes the assertion that any are fish," will surely come with out flaw from the premises. It certainly is more than a bare denial, and it is no mere repetition of the premises. And to say, If A does not exclude C after the middle has been elided, there shall be no inference and there can be no conclusion, seems purely arbitrary. Nor indeed do I see how this in sistence on elision, if we pressed it to its consequences, would prove compatible with the general validity of the third figure.
§ 9. The result we are left with may thus be stated. From CHAP. V NEGATIVE REASONING 283