The Principles of Mathematics (1903)

§ 70

A class, as we have seen, is neither a predicate nor a class-concept, for different predicates and different class-concepts may correspond to the same class. A class also, in one sense at least, is distinct from the whole composed of its terms, for the latter is only and essentially one, while the former, where it has many terms, is, as we shall see later, the very kind of object of which many is to be asserted. The distinction of a class as many from a class as a whole is often made by language: space and points, time and instants, the army and the soldiers, the navy and the sailors, the Cabinet and the Cabinet Ministers, all illustrate the distinction. The notion of a whole, in the sense of a pure aggregate which is here relevant, is, we shall find, not always applicable where the notion of the class as many applies (see Chapter X). In such cases, though terms may be said to belong to the class, the class must not be treated as itself a single logical subject[56]. But this case never arises where a class can be generated by a predicate. Thus we may for the present dismiss this complication from our minds. In a class as many, the component terms, though they have some kind of unity, have less than is required for a whole. They have, in fact, just so much unity as is required to make them many, and not enough to prevent them from remaining many. A further reason for distinguishing wholes from classes as many is that a class as one may be one of the terms of itself as many, as in classes are one among classes (the extensional equivalent of class is a class-concept), whereas a complex whole can never be one of its own constituents.(§ 70 ¶ 1)

§ 70 n. 1. A plurality of terms is not the logical subject when a number is asserted of it: such propositions have not one subject, but many subjects. See end of § 74.