The Principles of Mathematics (1903)

§ 36

The connected notions of the null-class and the existence of a class are next dealt with. In the edition of 1897, a class is defined as null when it is contained in every class. When we remember the definition of one class a being contained in another b (x is an a implies x is a b for all values of x), we see that we are to regard the implication as holding for all values, and not only for those values for which x really is an a. This is a point upon which Peano is not explicit, and I doubt whether he has made up his mind on it. If the implication were only to hold when x really is an a, it would not give a definition of the null-class, for which this hypothesis is false for all values of x. I do not know whether it is for this reason or for some other that Peano has since abandoned the definition of the inclusion of classes by means of formal implication between propositional functions: the inclusion of classes appears now to be regarded as indefinable. Another definition which Peano has sometimes favoured (e.g. F. 1895, Errata, p. 116) is, that the null-class is the product of any class into its negation—a definition to which similar remarks apply. In R. d. M. VII, No. 1 (§ 3, Prop. 1. 0), the null-class is defined as the class of those terms that belong to every class, i.e. the class of terms x such that a is a class implies x is an a for all values of a. There are of course no such terms x; and there is a grave logical difficulty in trying to interpret extensionally a class which has no extension. This point is one to which I shall return in Chapter VI.(§ 36 ¶ 1)

From this point onward, Peano’s logic proceeds by a smooth development. But in one respect it is still defective: it does not recognize as ultimate relational propositions not asserting membership of a class. For this reason, the definitions of a function[33] and of other essentially relational notions are defective. But this defect is easily remedied by applying, in the manner explained above, the principles of the Formulaire to the logic of relations[34].(§ 36 ¶ 2)

§ 36 n. 1. E.g. F. 1901, Part I, § 10, Props. 1. 0. 01 (p. 33).

§ 36 n. 2. See my article Sur la logique des relations, R. d. M. Vol. VII, 2 (1901).