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'.