§ 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 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.
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 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!
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 two denials there is no conclusion. If one premise denies and keeps to denial, then one premise at least is limited to the genus of subject and attribute. If the middle term B falls out of the conclusion, if A and C are connected through B, but not by means of an intermediate B, then the conclusion denies and falls also within the above-named genus. But if B is kept standing, the conclusion may at least in part be positive, and is not confined to a single category.
The general formula for negative reasoning, if we confine ourselves to the side of bare denial, may be stated as follows: 5 If B repels a content C, and is in relation with a third term A, then A and C will either be related directly by way of denial or else will be elements in a whole A — B — C, of which at least one member will be confined to the genus of subject and attribute. And I think with this we may take leave of a subject which has proved perhaps more troublesome than in teresting.
1 The statement that all reasoning, negative as well as positive, depends on an ideal whole, and that this whole can be called a con struction, is so far correct. But otherwise this section, and much of what follows, is unsatisfactory. Every negation (see on Bk. I. Chap. III.) implies a disjunction. And only because, and so far as, negative reasoning is based on and further developes a disjunctive totality and system— does it possess a real value. For an admirable exposition of this view the reader is referred to Bosanquet's Logic.
If we keep to mere denial, what is denied will certainly fall some where else in the Universe, since no mere ideas are possible. But, because the variety of special worlds within the Universe is indefinite, and because the merely denied is not, so far, located, you can base no special connection on the fact of mere simple denial. If negation is to be fruitful, it must (to repeat this) stand upon and move within a scheme of specialized alternatives, related to each other at once as positive and negative.
Hence it is scarcely worth while for me to attempt to correct chapter in detail. I will, however, touch on a certain number < points.
2 The usual demand for the elision of the middle term seems n defensible, and any rule that the conclusion must merely deny shou therefore be modified. See §8. But the rule which condemns two negative premisses, in the sense of two denials, must stand. For what is denied may fall in worlds not so connected as to make a construction possible. Hence, unless by going beyond mere denial one premiss becomes positive, no conclusion can be reached. In §4 after quaternio terminorum " we should add " or else one positive premiss."
3llf you keep to mere denial, as distinct from exclusion, repulsion or absence, all that is implied is an unspecified whole (x) containing two diversities (A and B). These must be positive, but, so far as you merely deny one of the other, you attend simply to their differ ence. Further, by identifying one of them (A) with C, you can deny the other (B) of C. But neither here nor elsewhere is there any inference through mere denial beyond the category of subject and attribute. As soon as you have assumed worlds containing arrangements and relations other than those of identity and difference, you have gone beyond mere negation in the sense of denial.
Hence the "general formula" (§9) can not stand, and should perhaps be read thus — " If you deny of B a content C, C can also be denied of that which is identical with B, and can further be related indirectly by denial with that which is related positively to B." But, though in the latter case the " conclusion " need not be " con fined to a single category," the inference, and what actually is concluded, never goes beyond the category of subject and attribute. Statements to the contrary (§§ 2, 8, and 9) are erroneous.
4 In the way of minor corrections I may here note that we should insert "definite" before "existence or position"; and (at the end of the paragraph) should read "move in any one real world at all." And, generally, I would remind the reader that such terms as "re moval," " exclusion," " repulsion," and even " absence," all are affirma tive in the sense of at least containing a positive aspect. And this aspect goes beyond what is contained in negation, if and so far as we take that as mere denial.
6 For " the general formula " see Note 3.
TWO CONDITIONS OF INFERENCE § I. We may briefly recapitulate the result we have reached. An inference is always an ideal construction result ing in the perception of a new connection. So far as this perception of the conclusion is concerned, there is no possibility of laying down rules, and the syllogistic logic teaches a super stition. That logic again has failed to include all the prin ciples of synthesis which operate in construction, and it is falsely confined to a single category. It is wrong again as to the number of the premises; and, in insisting on the neces sity of a major premise, it is clinging blindly to exploded meta physics in direct defiance of the most palpable facts. And it makes a further mistake as to the necessity of elision.
It might seem that having thus rejected the syllogism we must throw in our lot with its hereditary enemies. But yet, if the friends of the syllogism will allow it, we would rather take a place on their side. Our differences are trivial com pared with our agreements, and as against the enemy our cause is the same, for we have in common these two beliefs: (i) It is impossible to reason except upon the basis of identity, (ii) It is impossible to reason unless at least one premise is universal. It will be time to say vlcerunt empirici when these positions have both been forced.
§2. (i) I will begin with the necessity of an identical point. We know that an inference is an ideal construction, and the reality of this construction depends on its unity; if the construction is not individual it is merely fictitious. But how can any construction have unity unless it is united by a common point? And how can any point be common, unless in both the premises it is one and the same?
It is obvious that suppose the problem before us is to find the relation of S to P by means of their common relation to M, and if, by the hypothesis, S-M and M-P must be given separately, an advance is impossible, unless in both premises M is the same. Given S — M1 & M2 — P you can make no construction, for you have no bridge to carry you over from M1 to M2. The back of your inference now is broken and the extremities no longer belong to any individual principle. Un less M in both cases is absolutely the same you can not inter relate S and P.
If we are willing to give up the superstition of the copula and to admit a diversity of relations in judgment, we may say that in inference every pair of premises has one term the same, and that, if it is not the same, there can be no inference.
§ 3. It is obvious, if we dismiss our hardened prejudices and consider the question fairly by itself, that you can not argue on the strength of mere likeness.1 Whatever else may be right this at all events must be wrong; " A is similar to B, and B to C, and therefore A is like C," is a vicious infer ence, one that need not always be mistaken in fact, but that always must be a logical error. In practice I think we should all admit this. An inference based on nothing but likeness is utterly invalid; it is certainly ambiguous and probably false.
Likeness and sameness should never be confused, for the former refers properly to a general impression. Similarity is a perceived relation between two terms which implies and rests upon a partial identity. If we say that A and B are alike, we must be taken to assert that they have something the same. But we do not specify this point of sameness, and the moment we do that we have gone beyond mere similarity. If A and B for instance both have lungs or gills they are so far the same, and, on the strength of and because of this partial identity, they may present themselves to us as generally similar. But now add to these the further statement " B and C are alike." If we reduce the likeness here to partial identity we may find that the common point is here once again the possession of lungs or gills, and on the strength of this we may go on to argue that A and C (the extremes) are alike. But what actually interrelates A and C is not general similarity at all. If all you knew was that B was like C, the point of identity would be quite unspecified, and the fact might be, not that both had lungs or gills, but that each had one eye or the freedom of the will. In this case though each CHAP. VI TWO CONDITIONS OF INFERENCE 287 pair has its own internal likeness, you could not infer the similarity of A to C.
And if in answer I am told that this is irrelevant, and that it does not apply where the likeness is exact, I can only reply that I am waiting, and have been waiting for years, to be told what is meant by an " exact likeness." " A and B are not the same, but they are exactly alike, and therefore whatever is true of B must be true of A." But what can this mean? In the case of some twins it might be right to punish one for the other, and we should no longer care to identify criminals.
If a picture is " exactly like " a person, then if one is not dead the other will be alive. If a cast is " exactly like " an original I suppose the same thing will be in two places at once; and it is no mere metaphor if in certain cases the father is said to survive in his children, though the children might then cease to survive the father. But it is idle to pursue these frivolous consequences; the meaning which "exactly like" carries to my mind is nothing whatever but "partially the same" or "identical in some point or points." Likeness is always a perceived relation based upon a partial identity. In mere general similarity the identity will be indefinite; where the likeness is more special it must at least be partly defined, and where the similarity is called "exact" I understand that there is a definite point or points, in respect of which the same ness is complete. And if likeness did not imply identity all inference based upon it would be vicious. In practice every one would allow it to be vicious, nor do I understand how in theory it is possible to take it as having any other character.
I am most anxious to enter into (if I can), and to discuss the meaning our "advanced thinkers" may have attached " likeness " or " similarity." But I am forced to say again in this place what I had to say elsewhere some years aga While our "advanced thinkers" merely sing the old song which they have learnt and which their fathers have taught them, they can hardly expect to have its meaning discussed nor can they complain if they are treated as having no construction of given premises is not possible unless each pair of premises has a common point. And * Ethical Studies, p. 151 (Ed- IL P- 288 THE PRINCIPLES OF LOGIC BOOK II.?T. I common point must be an identical term. Thus in "A — B B — C therefore A — B — C" the B in each premise must not be merely alike, but must be absolutely the same. But here, after having avoided one error, we are threatened by another and opposite mistake. For if it is wrong to say that B is not the same, it is equally wrong to deny that it is different.
This may look mysterious but is really quite simple. If B in both premises were so far the same that no difference of any kind belonged to it, then it is obvious at once that both premises must be identical, or else that their differences do not concern B. But in each of these cases the inference dis appears. If the premises are the same their repetition is meaningless, and if the differences they contain are indifferent to B it is clear that no construction can be made, since, if B is the centre, it carries no radii and has no circumference. An identity which is not a synthesis of differences is plainly inert and utterly useless.