If the values of are the total values of a function fx for all values of x, then N() = ~(x).fx.
5.521 I separate the concept all from the truth-function.
Frege and Russell have introduced generality in connexion with the logical product of the logical sum. Then it would be difficult to understand the propositions "(x) . fx" e "(x) . fx" in which both ideas lie concealed.
5.522 That which is peculiar to the "symbolism of generality" is firstly, that it refers to a logical prototype, and secondly, that it makes constants prominent.
5.523 The generality symbol occurs as an argument.
5.524 If the objects are given, therewith are all objects also given.
If the elementary propositions are given, then therewith all elementary propositions are also given.
5.525 It is not correct to render the proposition "(x) . fx" - as Russell does - in the words "fx is possible".
Certainty, possibility or impossibility of a state of affairs are not expressed by a proposition but by the fact that an expression is a tautology, a significant proposition or a contradiction.
That precedent to which one would always appeal, must be present in the symbol itself.
5.526 (2) One can describe the world completely by completely generalized propositions, i.e. without from the outset co-ordinating any name with a definite object.
In order then to arrive at the customary way of expression we need simply say after an expression "there is only and only one x, which . . . .": and this x is a.