The Principles of Mathematics (1903)

§ 28

If R be a relation, we express by xRy the propositional function x has the relation R to y. We require a primitive (i.e. indemonstrable) proposition to the effect that xRy is a proposition for all values of x and y. We then have to consider the following classes: The class of terms which have the relation R to some term or other, which I call the class of referents with respect to R; and the class of terms to which some term has the relation R, which I call the class of relata. Thus if R be paternity, the referents will be fathers and the relata will be children. We have also to consider the corresponding classes with respect to particular terms or classes of terms; so-and-so’s children, or the children of Londoners, afford illustrations.(§ 28 ¶ 1)

The intensional view of relations here advocated leads to the result that two relations may have the same extension without being identical. Two relations R, R′ are said to be equal or equivalent, or to have the same extension, when xRy implies and is implied by xR′y for all values of x and y. But there is no need here of a primitive proposition, as there was in the case of classes, in order to obtain a relation which is determinate when the extension is determinate. We may replace a relation R by the logical sum or product of the class of relations equivalent to R, i.e. by the assertion of some or all such relations; and this is identical with the logical sum or product of the class of relations equivalent to R′, if R′ be equivalent to R. Here we use the identity of two classes, which results from the primitive proposition as to the identity of classes, to establish the identity of two relations—a procedure which could not have been applied to classes themselves without a vicious circle.(§ 28 ¶ 2)

A primitive proposition in regard to relations is that every relation has a converse, i.e. that, if R be any relation, there is a relation R′ such that xRy is equivalent to yR′x for all values of x and y. Following Schröder, I shall denote the converse of R by . Greater or less, before or after, implying and implied by, are mutually converse relations. With some relations, such as identity, diversity, equality, inequality, the converse is the same as the original relation; such relations are called symmetrical. When the converse is incompatible with the original relation, as in such cases as greater and less, I call the relation asymmetrical; in intermediate cases, not-symmetrical.(§ 28 ¶ 3)

The most important of the primitive propositions in this subject is that between any two terms there is a relation not holding between any two other terms. This is analogous to the principle that any term is the only member of some class; but whereas that could be proved, owing to the extensional view of classes, this principle, so far as I can discover, is incapable of proof. In this point, the extensional view of relations has an advantage; but the advantage appears to me to be outweighed by other considerations. When relations are considered intensionally, it may seem possible to doubt whether the above principle is true at all. It will, however, be generally admitted that, of any two terms, some propositional function is true which is not true of a certain given different pair of terms. If this be admitted, the above principle follows by considering the logical product of all the relations that hold between our first pair of terms. Thus the above principle may be replaced by the following, which is equivalent to it: If xRy implies x′Ry′, whatever R may be, so long as R is a relation, then x and x′ and y and y′ are respectively identical. But this principle introduces a logical difficulty from which we have been hitherto exempt, namely a variable with a restricted field; for unless R is a relation, xRy is not a proposition at all, true or false, and thus R, it would seem, cannot take all values, but only such as are relations. I shall return to the discussion of this point at a later stage.(§ 28 ¶ 4)