# The Principles of Mathematics (1903)

## § 126

In order to decide this point, it is necessary to pass to our third difficulty, and reconsider the notion of class itself. This notion appears to be connected with the notion of denoting, explained in Part I, Chapter V. We there pointed out five ways of denoting, one of which we called the numerical conjunction. This was the kind indicated by all. This kind of conjunction appears to be that which is relevant in the case of classes. For example, man being the class-concept, all men will be the class. But it will not be all men quâ concept which will be the class, but what this concept denotes, i.e. certain terms combined in the particular way indicated by all. The way of combination is essential, since any man or some man is plainly not the class, though either denotes combinations of precisely the same terms. It might seem as though, if we identify a class with the numerical conjunction of its terms, we must deny the distinction of a term from a class whose only member is that term. But we found in Chapter X that a class must always be an object of a different logical type from its members, and that, in order to avoid the proposition xx,this doctrine must be extended even to classes which have only one member. How far this forbids us to identify classes with numerical conjunctions, I do not profess to decide; in any case, the distinction between a term and the class whose only member it is must be made, and yet classes must be taken extensionally to the degree involved in their being determinate when their members are given. Such classes are called by Frege Werthverläufe; and cardinal numbers are to be regarded as classes in this sense.(§ 126 ¶ 1)