[PHI 3000] What did Kripke really show about identity?

It is often said that Kripke has shown that the notion of contingent identity is misguided. If a and b are identical (where a and b are rigid designators), then a and b are necessarily identical. The proof is supposed to go something like this:

1. Suppose that a = b.
2. a and b share all their properties in common [by Leibniz's law].
3. a has the property 'being necessarily identical with a'.
4. b has the property 'being necessarily identical with a' [from (2) & (3)]
5. b is necessarily identical with a [from (4)]

What do you make of this argument?

Kripke sets out to prove that true identity statements are necessary, not contingent. But premise 3 already assumes that true identity statements are necessary, i.e., that a is necessarily identical with a. In other words, to say that "a has the property 'being necessarily identical with a'" is just to say that "a is necessarily identical with a." But the argument sets out to prove that true identity statements are necessary. If that's the case, is the argument question-begging?

One might think that premise 3 is obviously true: a thing is necessarily identical to itself. But is it? What makes my computer identical to itself? What makes my car identical to itself? What makes me identical to myself?



This is, of course, the well-known problem of identity. If it is not clear what makes any thing identical to itself, is it still obvious that any thing is necessarily identical to itself?