The Principles of Mathematics (1903)

§ 74

A question which is very fundamental in the philosophy of Arithmetic must now be discussed in a more or less preliminary fashion. Is a class which has many terms to be regarded as itself one or many? Taking the class as equivalent simply to the numerical conjunction A and B and C and etc., it seems plain that it is many; yet it is quite necessary that we should be able to count classes as one each, and we do habitually speak of a class. Thus classes would seem to be one in one sense many in another.(§ 74 ¶ 1)

There is a certain temptation to identify the class as many and the class as one, e.g., all men and the human race. Nevertheless, wherever a class consists of more than one term, it can be proved that no such identification is permissible. A concept of a class, if it denotes a class as one, is not the same as any concept of the class which it denotes. That is to say, classes of all rational animals, which denotes the human race as one term, is different from men, which denotes men, i.e. the human race as many. But if the human race were identical with men, it would follow that whatever denotes the one must denote the other, and the above difference would be impossible. We might be tempted to infer that Peano's distinction, between a term and a class of which the said term is the only member, must be maintained, at least when the term in question is a class[59]. But it is more correct, I think, to infer an ultimate distinction between a class as many and a class as one, to hold that the many are only many, and are not also one. The class as one may be identified with the whole composed of the terms of the class, i.e., in the case of men, the class as one will be the human race.(§ 74 ¶ 2)

But can we now avoid the contradiction always to be feared, where there is something that cannot be made a logical subject? I do not myself see any way of eliciting a precise contradiction in this case. In the case of concepts, we were dealing with what was plainly one entity; in the present case, we are dealing with a complex essentially capable of analysis into units. In such a proposition as A and B are two, there is no logical subject: the assertion is not about A, nor about B, nor about the whole composed of both, but strictly and only about A and B. Thus it would seem that assertions are not necessarily about single subjects, but may also be about many subjects; and this removes the contradiction which arose, in the case of concepts, from the impossibility of making assertions about them unless they were turned into subjects. This impossibility being here absent, the contradiction which was to be feared does not arise.(§ 74 ¶ 3)

§ 74 n. 1. This conclusion is actually drawn by Frege from an analogous argument: Archiv für syst. Phil., 1, p. 444. See Appendix.