Apriority vs. Analyticity
It is often said, correctly I think, that there are contingent but a priori sentences, e.g. "water is the dominant liquid on earth". Are these sentences analytic or synthetic? That is, what puts you in a position to know these sentences? Does understanding suffice, or do you have to invoke some other a priori means, like Gödelian insight? To me this seems wildly and unnecessarily mysterious. Of course understanding suffices, at least in ordinary cases. So there are contingent but analytic sentences. I wonder why this is hardly ever said. Does anyone really believe that those statements are synthetic a priori?
The two-dimensional explanation of course applies here as well: Analyticity is necessity of A-intension, contingency contingency of C-intension. -- Haven't I just replaced 'apriority' by 'analyticity' in the two-dimensionalist account? I have. But isn't this what two-dimensionalists really meant? Suppose there are synthetic a priori truths. (If there are none, 'a priori' presumably coincides with 'analytic' -- unless some analytic truths are a posteriori -- and it doesn't matter how the account is stated.) The best candidates I can think of are mathematical truths. They certainly have a necessary C-intension, but I'm not so sure about the A-intension. Isn't both the truth and falsity of Goldbach's conjecture epistemically possible? Maybe not, because epistemic possibility is a matter of ideal rationality. But is it so clear that any ideally rational subject would be able to know the truth or falsity of Goldbach's conjecture? Of course, he will be able to derive it from second-order peano arithmetic. But on the assumption that arithmetical statements are synthetic, arithmetical truth can't consist in derivability from PA2, since (formal) derivability is an analytic matter. Moreover, what about undecidable sentences like the continuum hypothesis? Again, if mathematical truth is synthetic, it doesn't sound plausible to claim that CH is neither true nor false, since the only apparent reason for that claim is the undecidability in ZFC.
So unless every analytic truth is a priori, it is uncertain whether all a priori truths have a necessary A-intension. On the other hand, it is rather trivial that all analytic truths have a necessary A-intension.