The Principles of Mathematics (1903)

§ 485

By means of variable propositional functions, Frege obtains a definition of the relation which Peano calls ∈, namely the relation of a term to a class of which it is a member[126]. The definition is as follows: au is to mean the term (or the range of terms if there be none or many) x such that there is a propositional function ϕ which is such that u is the range of ϕ and ϕa is identical with x (p. 53). It is observed that this defines au whatever things a and u may be. In the first place, suppose u to be a range. Then there is at least one ϕ whose range is u, and any two whose range is u are regarded by Frege as identical. Thus we may speak of the function ϕ whose range is u. In this case, au is the proposition ϕa, which is true when a is a member of u, and is false otherwise. If, in the second place, u is not a range, then there is no such propositional function as ϕ, and therefore au is the range of a propositional function which is always false, i.e. the null-range. Thus au indicates the true when u is a range and a is a member of u; au indicates the false when u is a range and a is not a member of u; in other cases, au indicates the null-class.(§ 485 ¶ 1)

It is to be observed that from the equivalence of xu and xv for all values of x we can only infer the identity of u and v when u and v are ranges. When they are not ranges, the equivalence will always hold, since xu and xv are the null-range for all values of x; thus if we allowed the inference in this case, any two objects which are not ranges would be identical, which is absurd. One might be tempted to doubt whether u and v must be identical even when they are ranges: with an intensional view of classes, this becomes open to question.(§ 485 ¶ 2)

Frege proceeds (p. 55) to an analogous definition of the propositional function of three variables which I have symbolized as x R y, and here again he gives a definition which doesnot place any restrictions on the variability of R. This is done by introducing a double range, defined by a propositional function of two variables; we may regard this as a class of couples with sense[127]. If then R is such a class of couples, and if (x; y) is a member of this class class, x R y is to hold; in other cases it is to be false or null as before. On this basis, Frege successfully erects as much of the logic of relations as is required for his Arithmetic; and he is free from the restrictions on the variability of R which arise from the intentional view of relations adopted in the present work (cf. § 83).(§ 485 ¶ 3)

§ 485 n. 1. Cf. §§ 21, 76, supra.

§ 485 n. 2. Neglecting, for the present, our doubts as to there being any such entity as a couple with sense, cf. § 98.