(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.
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