In the Tractatus, Wittgenstein attempts a solution of Russells paradox
3.333 A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.
If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition “F(F(fx))”, and in this the outer function F and the inner function F must have different meanings;
for the inner has the form g(fx), the outer the form h(g(fx)).
Common to both functions is only the letter “F”, which by itself signifies nothing. This is at once clear, if instead of “F(F(u))” we write “There exists g : F(gu). gu = Fu”.
Herewith Russell’s paradox vanishes.
Does this work, or be made to work? (Assuming of course that this the same paradox that we now know by the name of Russell).
some defintions from Russells introduction:
From Russells introduction:
a. A propositional function is a function whose values are propositions; for example 'x is human'.
b. A truth-function of a proposition p is a proposition containing p and such that its truth or falsehood depends only upon the truth or falsehood of p.
c. Wittgenstein shows every propositional function is a truth-function.