The next set of distinctions belongs not to epistemology, but to metaphysics: necessity and contingency. At this point, it's no longer possible to dodge difficulties by appealing to the state of some particular ``knower'' or observer, here the question concerns truth and falsity independently of whether or not someone knows about it.
We ask whether something might have been true, or might have been false. Well, if something is false, it's obviously not necessarily true. If it is true, might it have been otherwise? Is it possible that, in this respect, the world should have been different from the way it is? If the answer is `no', then this fact about the world is a necessary one? If the answer is `yes', then this fact about the world is a contingent one. This in and of itself has nothing to do with anyone's knowledge of anything (36).
As I went over this passage this morning it occasioned several thoughts. In this passage, it's entirely unclear WHAT Kripke attaches `truth' to: is it propositions or facts? I believe it's the latter because this fits nicely with his essentialism. But it's entirely unclear to me that he distinguishes facts from propositions, which fits with my comment in connection with a priori that he's a strong realist as far as propositions are concerned. If propositions are indeed facts, a part of the furniture of the world, then saying that facts are true or false is quite ``natural.'' But if this is so, it becomes increasingly difficult to maintain the distinctions between metaphysics and epistemology that he wants to draw, the sharp demarcation between a priori and necessary disappears: if propositions are facts and can be assessed independently of any specific knower in order to ascertain their necessity, it seems entirely reasonable to say that they can be assessed for their potential to be known a priori, and if this is so it will be difficult to sustain the contingent a priori which is one important part of Kripke's claims. In turn, this will threaten his ``solution'' to the puzzles.
The passage gives us the definition of necessity: something is necessary iff it could not be otherwise; or if the world could not be such that it is false. For Kripke, it is ``not a matter of obvious definitional equivalence'' that the necessary is identical with the a priori. He uses the Goldbach conjecture (every even number greater than 2 is the sum of two primes) to illustrate this.
If the Goldbach conjecture is false, then there is an even number, n, greater than 2, such that for no primes p1 and p2, both <n, does n = p1 + p2. This fact about n, if true, is verifiable by direct computation, and thus is necessary if the results of arithmetical computations are necessary. On the other hand, if the conjecture is true, then every even number exceeding 2 is the sum of two primes. Could it then be the case that, although in fact every such even number is the sum of two primes, there might have been such an even number which was not the sum of two primes? (36)
Here we have a statement which NO ONE knows, whether a priori or not, but which is such that IF it is true, it's necessarily true (and likewise, if false, necessarily false). This seems persuasive, though I'm a bit bothered by his switching between talk of the truth of the conjecture itself to talk of specific instances of the conjecture. He next proceeds to eliminate one rejoinder.
Perhaps it will be alleged that we can in principle know a priori whether it is true. Well, maybe we can. Of courses an infinite mind which can search through all the numbers can or could. But I don't know whether a finite mind can or could. Maybe there is just no mathematical proof whatsoever which decides the conjecture (37).
Once again, this sounds fine, but is it? Remember how he defined a priori? ``whether a particular person or knower knows something a priori or believes it true on the basis of a priori evidence.'' The notion of evidence, what sort of evidence is appealed to is central to this. From this it follows that Kripke can identify a priori evidence; if this is so, there's no reason not to allow for the modal definition of a priori: p is a priori if there is a priori evidence which supports it. (Now that sounds circular, but if we replace the phrase ``a priori evidence'' with its definition it won't be. I leave it here because, unlike Kripke, I'm not sure what counts as a priori evidence.)
This impression is reinforced by his discussion of the Goldbach conjecture when he writes, ``This fact about n, if true, is verifiable by direct computation, and thus is necessary if the results of arithmetical computations are necessary.'' Maybe he means computation of an empirical sort: lining up groups of things and bunching them together to make sure they sum to the desired result, but I don't think so. This seems to be an instance of mustering a priori evidence if ever there was one.
But if all this is true, then indeed the a priori and the necessary do coincide, and coincide necessarily! Of course this is not to say that I may make a claim based on evidence which I gather empirically--for me it's a posteriori--but if it's a necessary truth then it's knowable a priori. This returns us to the earlier question I asked: If I say `p' based on the testimony of David Hilbert and Hilbert has proved `p' on the basis of a priori evidence, do I know the same thing Hilbert knows?