According to the view here advocated, it will be necessary, with every variable, to indicate whether its field of significance is terms, classes, classes of classes, or so on^{[134]}. A variable will not be able, except in special cases, to extend from one of these sets into another; and in `x`∈`u`, the `x` and the `u` must always belong to different types; ∈ will not be a relation between objects of the same type, but ∈∈̌ or ∈`R`∈̌^{[135]} will be, provided `R` is so. We shall have to distinguish also among relations according to the types to which their domains and converse domains belong; also variables whose fields include relations, these being understood as classes of couples, will not as a rule include anything else, and relations between relations will be different in type from relations between terms. This seems to give the truth--though in a thoroughly extensional form--underlying Frege's distinction between terms and the various kinds of functions. Moreover the opinion here advocated seems to adhere very closely indeed to common sense.(§ 492 ¶ 1)

Thus the final conclusion is, that the correct theory of classes is even more extensional than that of Chapter VI; that the class as many is the only object always defined by a propositional function, and that this is adequate for formal purposes; that the class as one, or the whole composed of the terms of the class, is probably a genuine entity except where the class is defined by a quadratic function (see § 103), but that in these cases, and in other cases possibly, the class as many is the only object uniquely defined.(§ 492 ¶ 2)

The theory that there are different kinds of variables demands a reform in the doctrine of formal implication. In a formal implication, the variable does not, in general, take all the values of which variables are susceptible, but only all those that make the propositional function in question a proposition. For other values of the variable, it must be held that any given propositional function becomes meaningless. Thus in `x`∈`u`, `u` must be a class, or a class of classes, or etc., and `x` must be a term if `u` is a class, a class if `u` is a class of classes, and so on; in every propositional function there will be some range permissible to the variable, but in general there will be possible values for other variables which are not admissible in the given case. This fact will require a certain modification of the principles of Symbolic Logic; but it remains true that, in a formal implication, all propositions belonging to a given propositional function are asserted.(§ 492 ¶ 3)

With this we come to the end of the more philosophical part of Frege's work. It remains to deal briefly with his Symbolic Logic and Arithmetic; but here I find myself in such complete agreement with him that it is hardly necessary to do more than acknowledge his discovery of propositions which, when I wrote, I believed to have been new.(§ 492 ¶ 4)

§ 492 n. 1. See Appendix B. ↩

§ 492 n. 2. On this notation, see §§ 28, 97. ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.