The Principles of Mathematics (1903)

§ 95

There is a certain temptation to affirm that no term can be related to itself; and there is a still stronger temptation to affirm that if a term can be related to itself, the relation must be symmetrical, i.e. identical with its converse. But both these temptations must be resisted. In the first place, if no term were related to itself, we should never be able to assert self-identity, since this is plainly a relation. But since there is such a notion as identity, and since it seems undeniable that every term is identical with itself, we must allow that a term may be related to itself. Identity, however, is still a symmetrical relation, and may be admitted without any great qualms. The matter becomes far worse when we have to admit not-symmetrical relations of terms to themselves. Nevertheless the following propositions seem undeniable; Being is, or has being; 1 is one, or has unity; concept is conceptual; term is a term; class-concept is a class-concept. All these are of one of the three equivalent types which we distinguished at the beginning of Chapter V, which may be called respectively subject-predicate propositions, propositions asserting the relation of predication, and propositions asserting membership of a class. What we have to consider is, then, the fact that a predicate may be predicable of itself. It is necessary, for our present purpose, to take our propositions in the second form (Socrates has humanity), since the subject-predicate form is not in the above sense relational. We may take, as the type of such propositions, unity has unity. Now it is certainly undeniable that the relation of predication is asymmetrical, since subjects cannot in general be predicated of their predicates. Thus unity has unity asserts one relation of unity to itself, and implies another, namely the converse relation: unity has to itself both the relation of subject to predicate, and the relation of predicate to subject. Now if the referent and the relatum are identical, it is plain that the relatum has to the referent the same relation as the referent has to the relatum. Hence if the converse of a relation in a particular case were defined by mutual implication in that particular case, it would appear that, in the present case, our relation has two converses, since two different relations of relatum to referent are implied by unity has unity. We must therefore define the converse of a relation by the fact that aRb implies and is implied by ba whatever a and b may be, and whether or not the relation R holds between them. That is to say, a and b are here essentially variables, and if we give them any constant value, we may find aRb implies and is implied by bR′a, where R′ is some relation other than .(§ 95 ¶ 1)

Thus three points must be noted with regard to relations of two terms: (1) they all have sense, so that, provided a and b are not identical, we can distinguish aRb from bRa; (2) they all have a converse, i.e. a relation such that aRb implies and is implied by ba, whatever a and b may be; (3) some relations hold between a term and itself, and such relations are not necessarily symmetrical, i.e. there may be two different relations, which are each other's converses, and which both hold between a term and itself.(§ 95 ¶ 2)