The Principles of Mathematics (1903)

§ 94

Next after subject-predicate propositions come two types of propositions which appear equally simple. These are the propositions in which a relation is asserted between two terms, and those in which two terms are said to be two. The latter class of propositions will be considered hereafter; the former must be considered at once. It has often been held that every proposition can be reduced to one of the subject-predicate type, but this view we shall, throughout the present work, find abundant reasons for rejecting. It might be held, however, that all propositions not of the subject-predicate type, and not asserting numbers, could be reduced to propositions containing two terms and a relation. This opinion would be more difficult to refute, but this too, we shall find, has no good grounds in its favour[66]. We may therefore allow that there are relations having more than two terms; but as these are more complex, it will be well to consider first such as have two terms only.(§ 94 ¶ 1)

A relation between two terms is a concept which occurs in a proposition in which there are two terms not occurring as concepts[67], and in which the interchange of the two terms gives a different proposition. This last mark is required to distinguish a relational proposition from one of the type a and b are two, which is identical with b and a are two. A relational proposition may be symbolized by aRB, where R is the relation and a and b are the terms; and aRb will then always, provided a and b are not identical, denote a different proposition from bRa. That is to say, it is characteristic of a relation of two terms that it proceeds, so to speak, from one to the other. This is what may be called the sense of the relation, and is, as we shall find, the source of order and series. It must be held as an axiom that aRb implies and is implied by a relational proposition bR′a, in which the relation R′ proceeds from b to a, and may or may not be the same relation as R. But even when aRb implies and is implied by bRa, it must be strictly maintained that these are different propositions. We may distinguish the term from which the relation proceeds as the referent, and the term to which it proceeds as the relatum. The sense of a relation is a fundamental notion, which is not capable of definition. The relation which holds between b and a whenever R holds between a and b will be called the converse of R, and will be denoted (following Schroder) by . The relation of R to is the relation of oppositeness, or difference of sense; and this must not be defined (as would seem at first sight legitimate) by the above mutual implication in any single case, but only by the fact of its holding for all cases in which the given relation occurs. The grounds for this view are derived from certain propositions in which terms are related to themselves not-symmetrically, i.e. by a relation whose converse is not identical with itself. These propositions must now be examined.(§ 94 ¶ 2)

§ 94 n. 2. This description, as we saw above (§ 48), excludes the pseudo-relation of subject to predicate.