u = F u ”.
Herewith Russell’s paradox vanishes.
The rules of logical syntax must follow of themselves, if we only know how every single sign signifies.
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.
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.
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.
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.)
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.
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.
What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.
We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, can be replaced by the notations of “ ~ p ” (“not p ”) and “ p ∨ q ” (“ p or q ”).
(Herewith is indicated the way in which a special possible notation can give us general information.)
The sign of the complex is not arbitrarily resolved in the analysis, in such a way that its resolution would be different in every propositional structure.
The proposition determines a place in logical space: the existence of this logical place is guaranteed by the existence of the constituent parts alone, by the existence of the significant proposition.