These are epistemological categories, but even among people who agree with this, there has been an inclination for them to suppose that this should be defined in terms of a modality: p is a priori iff p can be known independently of any experience (34). Quite apart from Quine's objections to this sort of move (see ``Two Dogmas''), Kripke sees a difficulty with this. If we say that p can be known independently of any experience, this means that it's possible, though not necessarily actual, that p be known independently of experience. But what is needed to make this true? If God knows p independently of experience, does this make it a priori? Well, presumably God knows EVERYTHING independently of experience, so that can't be it.
...For the Martian? Or just for people with minds like ours? To make this all clear might [involve] a host of problems all of its own about what sort of possibility is in question here. It might be best therefore, instead of using the phrase `a priori truth', to the extent that one uses it at all, to stick the question of whether a particular person or knower knows something a priori or believes it true on the basis of a priori evidence (34-35).
This seems to allow Kripke to avoid the Quinean intensional circle-though it really doesn't accomplish even this--and to side-step the uncomfortable position of holding that since David Hilbert knows some mathematical proposition a priori, it's just plain a priori and the rest of us either know it a priori or don't understand it.
However, there is a relativism that seems to have poked its nose in: what happens when Hilbert knows p a priori, and I know it based on the calculations of some computer calculation? do we know the same thing? what if I know it based on Hilbert's testimony? do I know even in this case what he knows? My intuitions aren't clear on this, but they incline toward the view that we don't know the same thing.
Even ignoring this, what Kripke says here is strange. He directs us to ``stick to the question whether a particular person or knower knows something a priori or believes it true on the basis of a priori evidence.'' This can only mean that S knows p a priori just in case S's evidence for p is independent of S's experience. He clearly wants to allow that while S may know p a priori, N may know it a posteriori.
Apart from its intrinsic interest, this raises another question: where
does this ``independent of experience'' evidence come from? This
takes us quite directly into the question of meaning and reference
fixing. If I fix the referent of `X' via 
then for me, `X is
' is a priori, though not, as we'll see, necessary
(unless it's an essential feature of X); the same may well not hold
for you. If, on the other hand, `X' means '', then `X is
' is necessary and a priori--well, for anyone who
knows what `X' means. Turning to the example of the question of
whether some very large number is prime, Kripke says
No one has calculated or proved that the number is prime; but the machine has given the answer: this number is prime. We, then, if we believe that the number is prime, believe it on the basis of our knowledge of the laws of physics, the construction of the machine, and so on. We therefore do not believe this on the basis of purely a priori evidence. We believe it (if anything is a posteriori at all) on the basis of a posteriori evidence. Nevertheless, maybe this could be known a priori by someone who made the requisite calculations. So `can be known a priori' doesn't mean `must be known a priori (35).
I find this a long rambling way around the issue. Could someone do the calculation? Suppose it took 1000 years to complete it? Even if we put a team of 1000 people on it, each would only do a fraction of the total calculation, would they all know it a priori? The most Kripke seems to have shown with the passage is that the notion of a priori, understood as he understands it, isn't especially interesting. But, wait.
Turning from the issues of who knows what, we might ask what this tells us about p. On the one hand, nothing since no proposition has an epistemic status until it is indexed to some knower. But it does tell us that Kripke believes that it is possible to talk about p independently of any knower, that propositions exist (in some sense) and have content/meaning whether or not they are in fact known by anyone. This may seem benign, but I suggest that it is part and parcel of Kripke's Referential Realism, and that it shows that he is deeply committed to the real existence of propositions and propositional content as trans-language entities. (We develop this argument in Chapter 15 of Word and World.) This saves a priori from being completely uninteresting, but the cost is rather high--it once again introduced metaphysics into epistemology, and it undercuts many of Kripke's own claims about meaning..