3.33

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In logical syntax the meaning of a sign ought never to play a role; 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 type").

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 functions 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.

3.334    The rules of logical syntax must follow of themselves, if we only know how every single sign signifies.