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:

is to mean the term (or the range of terms if there be none or many) `a`∈`u``x` such that there is a propositional function `ϕ` which is such that `u` is the range of `ϕ` and `ϕ``a` is identical with `x` (p. 53). It is observed that this defines `a`∈`u` 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, `a`∈`u` is the proposition `ϕ``a`, 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 `a`∈`u` is the range of a propositional function which is always false, i.e. the null-range. Thus `a`∈`u` indicates the true when `u` is a range and `a` is a member of `u`; `a`∈`u` indicates the false when `u` is a range and `a` is not a member of `u`; in other cases, `a`∈`u` indicates the null-class.(§ 485 ¶ 1)

It is to be observed that from the equivalence of `x`∈`u` and `x`∈`v` 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 `x`∈`u` and `x`∈`v` 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

, and here again he gives a definition which doesnot place any restrictions on the variability of `x` `R` `y``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 `(`

is a member of this class class, `x`; `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 `x` `R` `y``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. ↩

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