The Principles of Mathematics (1903)

§ 484

Classes. Frege's theory of classes is very difficult, and I am not sure that I have thoroughly understood it. He gives the name Werthverlauf[124] to an entity which appears to be nearly the same as what I call the class as one. The concept of the class, and the class as many, do not appear in his exposition. He differs from the theory set forth in Chapter VI chiefly by the fact that he adopts a more intensional view of classes than I have done, being led thereto mainly be the desirability of admitting hthe null-class and of distinguishing a term from a class whose only member it is. I agree entirely that hese two objects cannot be attained by an extensional theory, though I have tried to show how to satisfy the requirements of formalism (§§ 69, 73).(§ 484 ¶ 1)

The extension of a Begriff, Frege says, is the range of a function whose value for every argument is a truth-value (FuB. p. 16). Ranges are things, whereas functions are not (ib. p. 19). There would be no null-class, if classes were taken in extension; for the null-class is only possible if a class is not a collection of terms (KB. pp. 436-7). If x be a term, we cannot identify x, as the extensional view requires, with the class whose only member is x; for suppose x to be a class having more than one member, and let y, z be two different members of x; then if x is identical with the class whose only member is x, y and z will both be members of this class, and will therefore be identical with x and with each other, contrary to the hypothesis[125]. The extension of a Begriff has its being in the Begriff itself, not in the individuals falling under the Begriff (ib. p. 451). When I say something about all men, I say nothing about some wretch in the centre of Africa, who is in no way indicated, and does not belong to the indication of man (p. 454). Begriffe are prior to their extension, and it is a mistake to attempt, as Schröder does, to base extension on individuals; this leads to the calculus of regions (Gebiete), not to Logic (p. 455).(§ 484 ¶ 2)

What Frege understands by a range, and in what way it is to be conceived without reference to objects, he endeaovurs to explain in his Grundgesetze der Arithmetik. He begins by deciding that two propositional functions are to have the same range when they have the same value for every value of x, i.e. for every value of x both are true or both false (pp. 7, 14). This is laid down as a primitive proposition. But this only determines the equality of ranges, not what they are in themselves. If Χ(ξ) be a function which never has the same value for different values of ξ and if we denote by ϕ′ the range of ϕx, we shall have Χ(ϕ′) = Χ(ψ′) when and only when ϕ′ and ψ′ are equal, i.e. when and only when ϕx and ψx always have the same value. Thus the conditions for the equality of ranges do not of themselves decide what ranges are to be (p. 16). Let us decide arbitrarily--since the notion of a range is not yet fixed--that the true is to be the range of the function x is true (as an assumption, not an asserted proposition), and the false is to be the range of the function x=not every term is identical with itself. It follows that the range of ϕx is the true when and only when the true and nothing else falls under the Begriff ϕx; the range of ϕx is the false when and only when the false and nothing else falls under the Begriff ϕx; in other cases, the range is neither the true nor the false (pp. 17-18). If only one thing falls under a concept, this one thing is distinct from the range of the concept in question (p. 18, note)--the reason is the same as that mentioned above.(§ 484 ¶ 3)

There is an argument (p. 49) to prove that the name of the range of a function always has an indication, i.e. that the symbol employed for it is never meaningless. In view of the contradiction discussed in Chapter X, I should be inclined to deny a meaning to a range when we have a proposition of the form ϕ[f(ϕ)], where f is constant and ϕ variable, or of the form fx(x), where x is variable and fx is a propositional function which is determinate when x is given, but varies from one value of x to another--provided, when fx is analyzed into things and concepts, the part dependent on x does not consist only of things, but contains also at least one concept. This is a very complicated case, in which, I should say, there is no class as one, my only reason for saying so being that we can thus escape the contradiction.(§ 484 ¶ 4)

§ 484 n. 1. I shall translate this as range.

§ 484 n. 2. Ib. p. 444. Cf. supra, § 74.