The Principles of Mathematics (1903)

§ 79

To sum up the above somewhat lengthy discussion. A class, we agreed, is essentially to be interpreted in extension; it is either a single term, or that kind of combination of terms which is indicated when terms are connected by the word and. But practically, though not theoretically, this purely extensional method can only be applied to finite classes. All classes, whether finite or infinite, can be obtained as the objects denoted by the plurals of class-concepts--men, numbers, points, etc. Starting with predicates, we distinguished two kinds of proposition, typified by Socrates is human and Socrates has humanity, of which the first uses human as predicate, the second as a term of a relation. These two classes of propositions, though very important logically, are not so relevant to Mathematics as their derivatives. Starting from human, we distinguished (1) the class-concept man, which differs slightly, if at all, from human; (2) the various denoting concepts all men, every man, any man, a man, and some man; (3) the objects denoted by these concepts, of which the one denoted by all men was callled the class as many, so that all men (the concept) was called the concept of the class; (4) the class as one, i.e. the human race. We had also a classification of propositions about Socrates, dependent upon the above distinctions, and approximately parallel with them: (1) Socrates is-a man is nearly, if not quite, identical with Socrates has humanity; (2) Socrates is a-man expresses identity between Socrates and one of the terms denoted by a man; (3) Socrates is one among men, a proposition which raises difficulties owing to the plurality of men; (4) Socrates belongs to the human race, which alone expresses a relation of an individual to its class, and, as the possibility of relation requires, takes the class as one, not as many. We agreed that the null-class, which has no terms, is a fiction, though there are null class-concepts. It appeared throughout that, although any symbolic treatment must work largely with class-concepts and intension, classes and extension are logically more fundamental for the principles of Mathematics; and this may be regarded as our main general conclusion in the present chapter.(§ 79 ¶ 1)