Russell's Paradox

In 'theory of types' Russell had to consider the meaning of signs when establishing the rules for them. This is an error. We can dispose of Russell's paradox as follows: No proposition can make a statement about itself, because a propositional sign cannot contain itself. (That is the whole of the 'theory of types').

The reason why a function cannot be its own argument is that the function 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. In that case, we would have propositions such as: 'F(F(fx))' where the outer function F and the inner function F must have different meanings. This is because the inner one has the form ϕ(f(x)) and the outer one has the form ψ(ϕ(fx)).

Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(∃φ) : F(φu) . φu = Fu'.

The rules of logical syntax must be self evident, once one knows how each individual sign signifies.