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(Fu)” 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.
3.34
A proposition possesses essential and accidental features.
Accidental are the features which are due to a particular way of producing the propositional sign. Essential are those which alone enable the proposition to express its sense.
3.341
The essential in a proposition is therefore that which is common to all propositions which can express the same sense.
And in the same way in general the essential in a symbol is that which all symbols which can fulfill the same purpose have in common.
3.3411
One could therefore say the real name is that which all symbols, which signify an object, have in common. It would then follow, step by step, that no sort of composition was essential for a name.
3.342
In our notations there is indeed something arbitrary, but this is not arbitrary, namely that if we have determined anything arbitrarily, then something else must be the case. (This results from the essence of the notation.)
3.3421
A particular method of symbolizing may be unimportant, but it is always important that this is a possible method of symbolizing. And this happens as a rule in philosophy: The single thing proves over and over again to be unimportant, but the possibility of every single thing reveals something about the nature of the world.
3.343
Definitions are rules for the translation of one language into another. Every correct symbolism must be translatable into every other according to such rules. It is this which all have in common.