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.)

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 at all.

5.531 Thus I do not write 'f(a, b). a = b', but 'f(a, a)' (or 'f(b, b));
and not 'f(a,b). Pa = b', but 'f(a, b)'.

5.532 And analogously I do not write '(dx, y). f(x, y). x = y', but
'(dx) . f(x, x)'; and not '(dx, y). f(x, y). Px = y', but '(dx, y). f(x,
y)'. 5.5321 Thus, for example, instead of '(x): fx z x = a' we write
'(dx). fx . z: (dx, y). fx. fy'. And the proposition, 'Only one x
satisfies f( )', will read '(dx). fx: P(dx, y). fx. fy'.

5.533 The identity-sign, therefore, is not an essential constituent of
conceptual notation.

5.534 And now we see that in a correct conceptual notation
pseudo-propositions like 'a = a', 'a = b. b = c. z a = c', '(x). x = x',
'(dx). x = a', etc. cannot even be written down.

5.535 This also disposes of all the problems that were connected with
such pseudo-propositions. All the problems that Russell's 'axiom of
infinity' brings with it can