3.333 A function can therefore not be its own argument, because a functional sign already contains the prototype of its argument and it cannot contain itself.
Let us suppose that the function F(fx) could be its own argument; there would then therefore be a proposition: “F(F(fx))” and in this the outer function F and the inner function F must have different meanings, because the inner has the form ø(fx), the outer the form ψ(ø(fx)). Only the letter “F” is common to both functions, but that in itself has no significance.
This becomes clear immediately if instead of “F(F(u))” we write “(Eø) : F(øu) . øu = Fu”.
Thus Russell’s Paradox is laid to rest.
Black (p. 149) says of this reference to Russell’s paradox that Wittgenstein “presumably [means] the variant concerning functions (rather than classes) that are not themselves included in the set of their own values.”
Having rejected Russell’s solution in 3.332, here Wittgenstein rejects his problem too. (The “E” here should, by the way, be the existential quantifier.)
Ostrow (p. 67) compares this remark with 3.1432. “Wittgenstein’s aim is once more to bring out how our hold on a notion of logical form is parasitic on how we speak, on what it makes sense to say.” Russell’s belief that a theory of types is needed suggests that he confusedly thinks something like the opposite of this, Ostrow thinks.