5.52 If E has as its values all the values of a function fx for all values of x, then N(E) = P(dx) . fx.

5.521 I dissociate the concept all from truth-functions.  Frege and Russell introduced generality in association with logical productor logical sum.  This made it difficult to understand the propositions ‘(dx) . fx’ and ’(x) . fx’, in which both ideas are embedded.

5.522 What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants.

5.523 The generality-sign occurs as an argument.

5.524 If objects are given, then at the same time we are given all objects.  If elementary propositions are given, then at the same time all elementary propositions are given.

5.525 It is incorrect to render the proposition ‘(dx) . fx’ in the words, ’fx is possible ’ as Russell does.  The certainty, possibility, or impossibility of a situation is not expressed by a proposition, but by an expression’s being a tautology, a proposition with a sense, or a contradiction.  The precedent to which we are constantly inclined to appeal must reside in the symbol itself.

5.526 We can describe the world completely by means of fully generalized propositions, i.e. without first correlating any name with a particular object.

5.5261 A fully generalized proposition, like every other proposition, is composite. (This is shown by the fact that in ‘(dx, O) .  Ox’ we have to mention ‘O’ and ‘s’ separately.  They both, independently, stand in signifying relations to the world, just as is the case in ungeneralized propositions.) It is a mark of a composite symbol that it has something in common with other symbols.

5.5262 The truth or falsity of every proposition does make some alteration in the general construction of the world.  And the range that the totality of elementary propositions leaves open for its construction is exactly the same as that which is delimited by entirely general propositions. (If an elementary proposition is true, that means, at any rate, one more true elementary proposition.)

5.53 Identity of object I express by identity of sign, and not by using a sign for identity.  Difference of objects I express by difference of signs.

5.5301 It is self-evident that identity is not a relation between objects.  This becomes very clear if one considers, for example, the proposition ’(x) : fx . z . x = a’.  What this proposition says is simply that only a satisfies the function f, and not that only things that have a certain relation to a satisfy the function, Of course, it might then be said that only a did have this relation to a; but in order to express that, we should need the identity-sign itself.

5.5302 Russell’s definition of ‘=’ is inadequate, because according to it we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense .)