3.326 In order to recognize the symbol in the sign we must consider the significant use.
3.327 The sign determines a logical form only together with its logical syntactic application.
3.328 If a sign is not necessary then it is meaningless. That is the meaning of Occam’s razor.
(If everything in the symbolism works as though a sign had meaning, then it has meaning.)
3.33 In logical syntax the meaning of a sign ought never to play a rôle; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.
3.331 From this observation we get a further view—into Russell’s Theory of Types. Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.
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”).
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 ϕ(fx), the outer the form ψ(ϕ(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 “(∃ϕ):F(ϕu).ϕu=Fu”.
Herewith Russell’s paradox vanishes.
