# The Principles of Mathematics (1903)

## § 98

There is a temptation to regard a relation as definable in extension as a class of couples. This has the formal advantage that it avoids the necessity for the primitive proposition asserting that every couple has a relation holding between no other pair of terms. But it is necessary to give sense to the couple, to distinguish the referent from the relatum: thus a couple becomes essentially distinct from a class of two terms, and must itself be introduced as a primitive idea. It would seem, viewing the matter philosophically, that sense can only be derived from some relational proposition, and that the assertion that a is referent and b relatum already involves a purely relational proposition in which a and b are terms, though the relation asserted is only the general one of referent to relatum. There are, in fact, concepts such as greater, which occur otherwise than as terms in propositions having two terms (§§ 48, 54); and no doctrine of couples can evade such propositions. It seems therefore more correct to take an intensional view of relations, and to identify them rather with class-concepts than with classes. This procedure is formally more convenient, and also seems nearer to the logical facts. Throughout Mathematics there is the same rather curious relation of intensional and extensional points of view: the symbols other than variable terms (i.e. the variable class-concepts and relations) stand for intensions, while the actual objects dealt with are always extensions. Thus in the calculus of relations, it is classes of couples that are relevant, but the symbolism deals with them by means of relations. This is precisely similar to the state of things explained in relation to classes, and it seems unnecessary to repeat the explanations at length.(§ 98 ¶ 1)