The Principles of Mathematics (1903)

§ 488

(α) To either of these views there are grave objections. The former is the view of Frege and Peano. To realize the paradoxical nature of this view, it must be clearly grasped that it is not only the collection as many, but the collecti0on as one, that is distinct from the collection whose only term it is. (I speak of collections, because it is important to examine the bearing of Frege's argument upon the possibility of an extensional standpoint.) This view, in spite of its paradox, is certainly the one which seems to be required by the symbolism. It is quite essential that we should be able to regard a class as a single object, that there should be a null-class, and that a term should not (in general, at any rate) be identical with the class of which it is the only member. It is subject to these conditions that the symbolic meaning of class has to be interpreted. Frege's notion of a range may be identified with the collection as one, and all will then go well. But it is very hard to see any entity such as Frege's range, and the argument that there must be such an entity gives us little help. Moreover, in virtue of the contradiction, there certainly are cases where we have a collection as many, but no collection as one (§ 104). Let us then examine (β), and see whether this offers a better solution.(§ 488 ¶ 1)

(β) Let us suppose that a collection of one term is that one term, and that a collection of many terms is (or rather are) those many terms, so that there is not a single term at all which is the collection of the many terms in question. In this view there is, at first sight at any rate, nothing paradoxical, and it has the merit of admitting universally what the Contradiction shows to be sometimes the case. In this case, unless we abandon one of our fundamental dogmas, ∈ will have to be a relation of a term to its class-concept, not to its class; if a is a class-concept, what appears symbolically as the class whose only term is a will (one might suppose) be the class-concept under which falls only the concept a, which is of course (in general, if not always) different from a. We shall maintain, on account of the contradiction, that there is not always a class-concept for a given propositional function ϕx, i.e. that there is not always, for every ϕ, some class-concept a such that xa is equivalent to ϕx for all values of x; and the cases where there is no such class-concept will be cases in which ϕ is a quadratic form.(§ 488 ¶ 2)

So far, all goes well. But now we no longer have one definite entity which is determined equally by any one of a set of equivalent propositional functions, i.e. there is, it might be urged, no meaning of class left which is determined by extension alone. Thus, to take a case where this leads to confusion, if a and b be different class-concepts such that xa and xb are equivalent for all values of x, the class-concept under which a falls and nothing else will not be identical with that under which falls b and nothing else. Thus we cannot get any way of denoting what should symbolically correspond to the class as one. Or again, if u and v be similar but different classes, similar to u is a different concept from similar to v; thus, unless we can find some extensional meaning for class, we shall not be able to say that the number of u is the same as that of v. And all the usual elementary problems as to combinations (i.e. as to the number of classes of specified kinds contained in a given class) will have become impossible and even meaningless. For these various reasons, an objector might contend, something like the class as one must be maintained; and Frege's range fulfils the conditions required. It would seem necessary therefore to accept ranges by an act of faith, without waiting to see whether there are such things.(§ 488 ¶ 3)

Nevertheless, the non-identification of the class with the class as one, whether in my form or inthe form of Frege's range, appears unavoidable, and by a process of exclusion the class as many is left as the only object which can play the part of a class. By a modification of the logic hitherto advocated in the present work, we shall, I think, be able at once to satisfy the requirements of the Contradiction and to keep in harmony with common sense[128].(§ 488 ¶ 4)

§ 488 n. 1. The doctrine to be advocated in what follows is the direct denial of the dogma stated in § 70, note.