The Principles of Mathematics (1903)

§ 68

In Chapter II we regarded classes as derived from assertions, i.e. as all the entities satisfying some assertion, whose form was left wholly vague. I shall discuss this view critically in the next chapter; for the present, we may confine ourselves to classes as they are derived from predicates, leaving open the question whether every assertion is equivalent to a predication. We may, then, imagine a kind of genesis of classes, through the successive stages indicated by the typical propositions Socrates is human, Socrates has humanity, Socrates is a man, Socrates is one among men. Of these propositions, the last only, we should say, explicitly contains the class as a constituent; but every subject-predicate proposition gives rise to the other three equivalent propositions, and thus every predicate (provided it can be sometimes truly predicated) gives rise to a class. This is the genesis of classes from the intensional standpoint.(§ 68 ¶ 1)

On the other hand, when mathematicians deal with what they call a manifold, aggregate, Menge, ensemble, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class. Here it is not predicates and denoting that are relevant, but terms connected by the word and, in the sense in which this word stands for numerical conjunction. Thus Brown and Jones are a class, and Brown singly is a class. This is the extensional genesis of classes.(§ 68 ¶ 2)