Page:Wittgenstein - Tractatus Logico-Philosophicus, 1922.djvu/143

A characteristic of a composite symbol: it has something: in common with other symbols.

5.5262 The truth or falsehood of every proposition alters something in the general structure of the world. And the range which is allowed to its structure by the totality of elementary propositions is exactly that which the completely general propositions delimit.

(If an elementary proposition is true, then, at any rate, there is one more elementary proposition true.)

5.53 Identity of the object I express by identity of the sign and not by means of a sign of identity. Difference of the objects by difference of the signs.

5.5301 That identity is not a relation between objects is obvious. This becomes very clear if, for example, one considers the proposition "$(x):fx\supset .x=a$" What this proposition says is simply that only $a$ satisfies the function $f$, and not that only such things satisfy the function $f$ which have a certain relation to $a$.

One could of course say that in fact only $a$ has this relation to $a$ but in order to express this we should need the sign of identity itself.

.5302 Russell's definition of "$=$" won't do; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never true, it is nevertheless significant.)

5.5303 Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing.

5.531 I write therefore not "$f(a,b).a=b$" but "$f(a,a)$" (or "$f(b,b)$"). And not "$f(a,b).\sim a=b$" but "$f(a,b)$"

5.532 And analogously: not "$(\exists x,y).f(x,y).x=y$",