Does Wittgensteins own solution to Russells Paradox actually work?

Question

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.

Answer 1

I think that for an understanding Wittgenstein's "solution", you must take in account the original (historical) context.

Wittgenstein was a student and disciple of Frege and Russell.

According to Frege (see his immediate response to Russell's letter communicating to him the discovery of the Paradox), Russell's form of the Paradox ( f(f) ) was not reproducible in Frege's logic, because syntax rules prescribe that the name of a function has an empty place that can be fileld only by the name of an object, and functions (in Frege's logic) are not objects. Immediately after, Frege was able to reproduce the Paradox in his own logic, with a more "complicated" formulation.

According to Whitehead & Russell Principia Mathematica, the solution of the paradoxes (included Russell's one) was Ramified Type Theory. Thos theory was from the beginning subjected to several objections, included a difficulty connected with the formal exposition of the theory itself in PM (PM does not has a neat separation between object language and meta-language, and so no precise demarcation between object- and meta-theory).

One of Tractatus' proposed improvements upon Frege's and Russel's logics is the idea that logical form cannot be described: it can only be showed. A plausible reading of this idea his that syntax rules are already enclosed into the signs themselves : the possibility of hanging together different signs by itself prevents paradoxical constructions (like f(f) ).

Answer 2

Wittgenstein is alluding to how Russell himself solved the Paradox - the theory of ramified types. He alludes to this in:

3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).

And he reformulates as

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.

A functional sign is simply the sign of the function; the function being what the sign signifies. He expands what he means by this:

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)).

That is F(F(fx)) is different from F(F(fx)) because in the expression they signify different things, that is they have different meanings or precisely functions; and only the sign 'F' is common to both, as he affirms:

Common to both functions is only the letter “F”, which by itself signifies nothing.

and by

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.