ABSTRACT
A relation between two terms is a concept which occurs in a proposition in which there are two terms not occurring as concepts,† 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 Schröder) by R˘. The relation of R to R˘ 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 notsymmetrically, i.e. by a relation whose converse is not identical with itself. These propositions must now be examined.
