The Principles of Mathematics (1903)

§ 57

The notion of denoting may be obtained by a kind of logical genesis from subject-predicate propositions, upon which it seems more or less dependent. The simplest of propositions are those in which one predicate occurs otherwise than as a term, and there is only one term of which the predicate in question is asserted. Instances are: A is, A is one, A is human. Concepts which are predicates might also be called class-concepts, because they give rise to classes, but we shall find it necessary to distinguish between the words predicate and class-concept. Propositions of the subject-predicate type always imply and are implied by other propositions of the type which asserts that an individual belongs to a class. Thus the above instances are equivalent to: A is an entity, A is a unit, A is a man. These new propositions are not identical with the previous ones, since they have an entirely different form. To begin with, is is now the only concept not used as a term. A man, we shall find, is neither a concept nor a term, but a certain kind of combination of certain terms, namely of those which are human. And the relation of Socrates to a man is quite different from his relation to humanity; indeed Socrates is human must be held, if the above view is correct, to be not, in the most usual sense, a judgment of relation between Socrates and humanity, since this view would make human occur as term in Socrates is human. It is, of course, undeniable that a relation to humanity is implied by Socrates is human, namely the relation expressed by Socrates has humanity; and this relation conversely implies the subject-predicate proposition. But the two propositions can be clearly distinguished, and it is important to the theory of classes that this should be done. Thus we have, in the case of every predicate, three types of propositions which imply one another, namely, Socrates is human, Socrates has humanity, and Socrates is a man. The first contains a term and a predicate, the second two terms and a relation (the second term being identical with the predicate of the first proposition)[47], while the third contains a term, a relation, and what I shall call a disjunction (a term which will be explained shortly)[48]. The class-concept differs little, if at all, from the predicate, while the class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate. The relation which occurs in the second type (Socrates has humanity) is characterized completely by the fact that it implies and is implied by a proposition with only one term, in which the other term of the relation has become a predicate. A class is a certain combination of terms, a class-concept is closely akin to a predicate, and the terms whose combination forms the class are determined by the class-concept. Predicates are, in a certain sense, the simplest type of concepts, since they occur in the simplest type of proposition.(§ 57 ¶ 1)

§ 57 n. 2. There are two allied propositions expressed by the same words, namely Socrates is a-man and Socrates is-a man. The above remarks apply to the former; but in future, unless the contrary is indicated by a hyphen or otherwise, the latter will always be in question. The former expresses the identity of Socrates with an ambiguous individual; the latter expresses a relation of Socrates to the class-concept man.