The Principles of Mathematics (1903)

§ 157

The decision between the absolute and relative theories can be made at once by appealing to a certain general principle, of very wide application, which I propose to call the principle of Abstraction. This principle asserts that, whenever a relation, of which there are instances, has the two properties of being symmetrical and transitive, then the then the relation in question is not primitive, but is analyzable into sameness of relation to some other term; and that this common relation is such that there is only one term at most to which a given term can be so related, though many terms may be so related to a given term. (That is, the relation is like that of son to father: a man may have many sons, but can have only one father.(§ 157 ¶ 1)

This principle, which we have already met with in connection with cardinals, may seem somewhat elaborate. It is, however, capable of proof, and is merely a careful statement of a very common assumption. It is generally held that all relations are analyzable into identity or diversity of content. Though I entirely reject this view, I retain, so far as symmetrical transitive relations are concerned, what is really a somewhat modified statement of the traditional doctrine. Such relations, to adopt more usual phraseology, are always constituted by possession of a common property. But a common property is not a very precise conception, and will not, in most of its ordinary significations, formally fulfil the function of analyzing the relations in question. A common quality of two terms is usually regarded as a predicate of those terms. But the whole doctrine of subject and predicate, as the only form of which propositions are capable, and the whole denial of the ultimate reality of relations, are rejected by the logic advocated in the present work. Abandoning the term predicate, we may say that the most general sense which can be given to a common property is this: A common property of two terms is any third term to which both have one and the same relation. In this general sense, the possession of a common property is symmetrical, but not necessarily transitive. In order that it may be transitive, the relation to the common property must be such that only one term at most can be the property of any given term[107]. Such is the relation of a quantity to its magnitude, or of an event to the time at which it occurs: given one term of the relation, namely the referent, the other is determinate, but given the other, the one is by no means determinate. Thus it is capable of demonstration that the possession of a common property of the type in question always leads to a symmetrical transitive relation. What the principle of abstraction asserts is the converse, that such relations only spring from common properties of the above type[108]. It should be observed that the relation of the terms to what I have called their common property can never be that which is usually indicated by the relation of subject to predicate, or of the individual to its class. For no subject ( in the received view) can have only one predicate, and no individual can belong to only one class. The relation of the terms to their common property is, in general, different in different cases. In the present case, the quantity is a complex of which the magnitude forms an element: the relation of the quantity to the magnitude is further defined by the fact that the magnitude has to belong to a certain class, namely that of magnitudes. It must then be taken as an axiom (as in the case of colours) that two magnitudes of the same kind cannot coexist in one spatio-temporal place, or subsist as relations between the same pair of terms; and this supplies the required uniqueness of the magnitude. It is such synthetic judgments of incompatibility that lead to negative judgments; but this is a purely logical topic, upon which it is not necessary to enlarge in this connection.(§ 157 ¶ 2)

§ 157 n. 1. The proof of these assertions is mathematical, and depends upon the Logic of Relations; it will be found in my article Sur la Logique des Relations, R. d. M. VII, No. 2, § 1, Propos. 6. 1, and 6. 2.

§ 157 n. 2. The principle is proved by showing that, if R be a symmetrical transitive relation, and a a term of the field of R, a has, to the class of terms to which it has the relation R taken as a whole, a many-one relation which, relationally multiplied by its converse, is equal to R. Thus a magnitude may, so far as formal arguments are concerned, be identified with a class of equal quantities.