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 = F u”. Herewith Russell’s paradox vanishes.

I fairly understand what the author means here, but it is in the bold part that I'm having difficult understanding what is meant. I don't understand why the function φ should be unique in this case, and how it relates to Russel's paradox.

  • @ClydeFrog Yes these answers were helpful and I understand it now. It is going to sound dumb, but I must be honest, I kept interpreting the period as a multiplier symbol and that completely threw me off. – Weezy Aug 4 '20 at 19:59
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    Yeah, you gotta get into this notation. Have fun with Wittgenstein! – Mr. White Aug 4 '20 at 20:12

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