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 .. 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.
5.5302 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, to say of one thing that it is identical with itself is to say nothing.