How is Saul Kripke said to have revive metaphysics in analytic philosophy tradition again?
Because he provides a logical framework within which to understand what it means to say that a thing has some property necessarily. Call this view essentialism.
Quine had thought that essentialism was false because necessity could only be a property of sentences. (So there could be necessary truths, but not necessary properties of objects.) Quine's reason for saying this was that there are logical paradoxes that appear to arise if we attribute necessary properties to objects. For instance:
- The number 8 is necessarily even.
- The number of the planets is 8.
- Therefore the number of the planets is necessarily even.
But 3 is false. Proponents of essentialism would regard (3) as a paradigm case of a false sentence. They want to say that the fact that we have 8 planets in the solar system rather than 7, or whatever is just a contingent fact.
What Kripke's semantics for modal logic gives us is the ability to make a distinction between:
3*. It is necessarily true that the number of planets is even. 3**. It is necessarily true of the number of planets, that that number is even.
The difference being that 3*, but not 3** implies 3. Therefore, Quine's objection doesn't go through.
This is all still controversial, but this is the guiding idea behind why Kripke is so important for contemporary metaphysics.
Following on from Shanes answer; and making the connection with the world from a logical analysis of language; I'd also look at the semantics of Kripkes modal logic and its interpreatation: that is something is neccessarily true iff it is true in every possible world.
I can conceive of a world where the number of planets in our solar system is 5; thus it is not neccesarily true that the number of planets in our solar system in 8.
But I can't conceive of any possible world where 8 is not even; thus this is a neccessary truth.