# The Principles of Mathematics (1903)

## § 76

Something must be said as to the relation of a term to a class of which it is a member, and as to the various allied relations. One of the allied relations is to be called , and is to be fundamental in Symbolic Logic. But it is to some extent optional which of them we take as symbolically fundamental.(§ 76 ¶ 1)

Logically, the fundamental relation is that of subject and predicate, expressed in Socrates is human--a relation which, as we saw in Chapter IV, is peculiar in that the relatum cannot be regarded as a term in the proposition. The first relation that grows out of this is the one expressed by Socrates has humanity, which is distinguished by the fact that here the relation is a term. Next comes Socrates is a man. This proposition, considered as a relation between Socrates and the concept man, is the one which Peano regards as fundamental; and his expresses the relation is a between Socrates and man. So long as we use class-concepts for classes in our symbolism, this practice is unobjectionable; but if we give this meaning, we must not assume that two symbols representing equal class-concepts both represent one and the same entity. We may go on to the relation between Socrates and the human race, i.e. between a term and its class considered as a whole; this is expressed by Socrates belongs to the human race. This relation might equally well be represented by . It is plain that, since a class, except when it has one term, is essentially many, it cannot be as such represented by a single letter: hence in any possible Symbolic Logic the letters which do duty for classes cannot represent the classes as many, but must represent either class-concepts, or the wholes composed of classes, or some other allied single entities. And thus cannot represent the relation of a term to its class as many; for this would be a relation of one term to many terms, not a two-term relation such as we want. This relation might be expressed by Socrates is one among men; but this, in any case, cannot be taken to be the meaning of .(§ 76 ¶ 2)