The Principles of Mathematics (1903)

§ 55

It may be doubted whether the general concept difference occurs at all in the proposition A differs from B, or whether there is not rather a specific difference of A and B, and another specific difference of C and D, which are respectively affirmed in A differs from B and C differs from D. In this way, difference becomes a class-concept of which there are as many instances as there are pairs of different terms, and the instances may be said, in Platonic phrase, to partake of the nature of difference. As this point s quite vital in the theory of relations, it may be well to dwell upon it. And first of all, I must point out that in A differs from B I intend to consider the bare numerical difference in virtue of which they are two, not difference in this or that respect.(§ 55 ¶ 1)

Let us first try the hypothesis that a difference is a complex notion, compounded of difference together with some special quality distinguishing a particular difference from every other particular difference. So far as the relation of difference itself is concerned, we are to suppose that no distinction can be made between different cases; but there are to be different associated qualities in different cases. But since cases are distinguished by their terms, the quality must be primarily associated with the terms, not with difference. If the quality be not a relation, it can have no special connection with the difference of A and B, which it was to render distinguishable from bare difference, and if it fails in this it becomes irrelevant. On the other hand, if it be a new relation between A and B, over and above difference, we shall have to hold that any two terms have two relations, difference and a specific difference, the latter not holding between any other pair of terms. This view is a combination of two others, of which the first holds that the abstract general relation of difference itself holds between A and B, while the second holds that when two terms differ they have, corresponding to this fact, a specific relation of difference, unique and unanalyzable and not shared by any other pair of terms. Either of these views may be held with either the denial or the affirmation of the other. Let us see what is to be said for and against them.(§ 55 ¶ 2)

Against the notion of specific differences, it may be urged that, if differences differ, their differences from each other must also differ, and thus we are led into an endless process. Those who object to endless processes will see in this a proof that differences do not differ. But in the present work, it will be maintained that there are no contradictions peculiar to the notion of infinity, and that an endless process is not to be objected to unless it arises in the analysis of the actual meaning of a proposition. In the present case, the process is one of implications, not one of analysis; it must therefore be regarded as harmless.(§ 55 ¶ 3)

Against the notion that the abstract relation of difference holds between A and B, we have the argument derived from the analysis of A differs from B, which gave rise to the present discussion. It is to be observed that the hypothesis which combines the general and the specific difference must suppose that there are two distinct propositions, the one affirming the general, the other the specific difference. Thus if there cannot be a general difference between A and B, this mediating hypothesis is also impossible. And we saw that the attempt to avoid the failure of analysis by including in the meaning of A differs from B the relations of difference to A and B was vain. This attempt, in fact, leads to an endless process of the inadmissible kind; for we shall have to include the relations of the said relations to A and B and difference, and so on, and in this continually increasing complexity we are supposed to be only analyzing the meaning of our original proposition. This argument establishes a point of very great importance, namely, that when a relation holds between two terms, the relations of the relation to the terms, and of these relations to the relation and the terms, and so on ad infinitum, though all implied by the propositions affirming the original relation, form no part of the meaning of this proposition.(§ 55 ¶ 4)

But the above argument does not suffice to prove that the relation of A to B cannot be abstract difference: it remains tenable that, as was suggested to begin with, the true solution lies in regarding every proposition as having a kind of unity which analysis cannot preserve, and which is lost even though it be mentioned by analysis as an element of the proposition. This view has doubtless its own difficulties, but the view that no two pairs of terms can have the same relation both contains difficulties of its own and fails to solve the difficulty for the sake of which it was invented. For, even if the difference of A and B be absolutly peculiar to A and B, still the three terms A, B, difference of A from B, do not reconstitute the proposition A differs from B, any more than A and B and difference did. And it seems plain that, even if differences did differ, they would still have to have something in common. But the most general way in which two terms can have something in common is by both having a given relation to a given term. Hence if no two pairs of terms can have the same relation, it follows that no two terms can have anything in common, and hence different differences will not be in any definable sense instances of difference[46]. I conclude, then, that the relation affirmed between A and B in the proposition A differs from B is the general relation of difference, and is precisely and numerically the same as the relation affirmed between C and D in C differs from D. And this doctrine must be held, for the same reasons, to be true of all other relations; relations do not have instances, but are strictly the same in all propositions to which they occur.(§ 55 ¶ 5)

We may now sum up the main poins elicited in our discussion of the verb. The verb, we saw, is a concept which, like the adjective, may occur in a proposition without being one of the terms of the proposition, though it may also be made into a logical subject. One verb, and one only, must occur as verb in every proposition; but every proposition, by turning its verb into a verbal noun, can be changed into a single logical subject, of a kind which I shall call in the future a propositional concept. Every verb, in the logical sense of the word, may be regarded as a relation; when it occurs as verb, it actually relates, but when it occurs as verbal noun it is the bare relation considered independently of the term which it relates. Verbs do not, like adjectives, have instances, but are identical in all the cases of their occurrence. Owing to the way in which the verb actually relates the terms of a proposition, every proposition has a unity which renders it distinct from the sum of its constituents. All these points lead to logical problems, which, in a treatise on logic, would deserve to be fully and thoroughly discussed.(§ 55 ¶ 6)

Having now given a general sketch of the nature of verbs and adjectives, I shall proceed, in the next two chapters, to discussions arising out of the consideration of adjectives, and in Chapter VII to topics connected with verbs. Broadly speaking, classes are connected with adjectives, while propositional functions involve verbs. It is for this reason that it has been necessary to deal at such length with a subject which might seem, at first sight, to be somewhat remote from the principles of mathematics.(§ 55 ¶ 7)

§ 55 n. 1. The above argument appears to prove that Mr Moore's theory of universals with numerically diverse instances in his paper on Identity (Proceedings of the Aristotelian Society, 1900--1901) must not be applied to all concepts. The relation of an instance to its universal, at any rate, must be actually and numerically the same in all cases where it occurs.