The Principles of Mathematics (1903)

§ 21

The insistance on the distinction between ∈ and the relation of whole and part between classes is due to Peano, and is of very great importance to the whole technical development and the whole of the applications to mathematics. In the scholastic doctrine of the syllogism, and in all previous symbolic logic, the two relations are confounded, except in the work of Frege[17]. The distinction is the same as that between the relation of individual to species and that of species to genus, between the relation of Socrates to the class of Greeks and the relation of Greeks to men. On the philosophical nature of this distinction I shall enlarge when I come to deal critically with the nature of classes; for the present it is enough to observe that the relation of whole and part is transitive, while ∈ is not so: we have Socrates is a man, and men are a class, but not Socrates is a class. It is to be observed that the class must be distinguished from the class-concept or predicate by which it is to be defined: thus men are a class, while man is a class-concept. The relation ∈ must be regarded as holding between Socrates and men considered collectively, not between Socrates and man. I shall return to this point in Chapter VI. Peano holds that all propositional functions containing only a single variable are capable of expression in the form x is an a, where a is a constant class; but this view we shall find reason to doubt.(§ 21 ¶ 1)

§ 21 n. 1. See his Begriffsschrift, Halle, 1879, and Grundgesetze der Arithmetik, Jena, 1893, p. 2.