In My Philosophical Development Russell wrote,
I come next to what Wittgenstein had to say about identity, which has an importance that may not be obvious at once. To explain this theory, I must first say something about the definition of identity in Principia Mathematica. Among the properties that an object may have, Whitehead and I distinguished some as what we called 'predicative'. These were properties which did not refer to any totality of properties. You may say, for instance,'Napoleon was Corsican' ..., and, in saying such things, you do not refer to any assemblage of properties. But if you say 'Napoleon had all the qualities of a great general' or 'Queen Elizabeth I had all the virtues of her father and grandfather and the vices of neither' you are referring to a totality of qualities. Properties that in this way refer to a totality we distinguished from predicative functions in order to avoid certain contradictions. We defined 'x is identical with y' as meaning 'y has all the predicative properties of x', and, in our system, it followed that y had any property that x had, whether predicative or not. To this, Wittgenstein objected as follows: '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.)
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 (Tractatus, 5.5302 and 5.5303)
At one time I accepted this criticism, but I soon came to the conclusion that it made mathematical logic impossible and, in fact, that Wittgenstein's criticism is invalid. This appears especially if we consider counting: if a and b have all their properties in common, you can never mention a without mentioning b or count a without at the same time counting b, not as a separate item but in the same act of counting. You could, therefore, never conceivably discover that a and b were two. Wittgenstein's position assumes that diversity is an indefinable relation, although I do not think that he knew he was making this assumption. But if he is not making it, I do not see on what grounds he can say, as he does, that it is significant, to say that two objects have all their properties in common. If, however, diversity is admitted, then, if a and b are two, a has a property which b has not, namely, that of being diverse from b. I think, therefore, that Wittgenstein's contention as to identity is mistaken. And, if so, it invalidates a large part of his system.
I don't understand how Russell's argument (second paragraph) refutes that of Wittgenstein's. Even in view of Russell's arguments, I think Wittgenstein's arguments make sense. One may or may not "conceivably discover" that a and b are distinct, but that doesn't alter the fact that in fact they are (if it's possible at all).
Also, it seems to me that the definition of "=" could be made more precise if we would rephrase it as,
x=y iff y has all the predicative properties of x and conversely.
If this is granted then I think that Wittgenstein's argument makes even more sense. If for example there were two things x and y such that, x=y then we can say that x and y are same object, i.e., there are in fact not two things but only one and it is surely nothing significant to say that two identical things are identical.
Can anyone help me in explaining it?