The nature of a priori evidence has gotten a lot of attention of late. Broadly speaking, a rift of sorts has developed between those who think that a priori evidence should be analyzed in terms of conceivability or modal imagination (e.g., Yablo and Chalmers) and those who think that it should be analyzed in terms of a distinct sui generis mental faculty of intuition (e.g., Bealer). The rift is of general philosophical interest because only the former approach appears to be compatible with a neo-empiricist epistemology.
It seems to me, however, that the conceivability camp cannot entirely eliminate intuition in favor of modal imagination. To see the problem, consider Dave Chalmer's account of positive conceivability:
[T]o positively conceive of a situation is to imagine (in some sense) a specific configuration of objects and properties. ... . When one imagines a situation and reasons about it, the object of one's imagination is often revealed as a situation in which S is the case, for some S. When this is so, we can say that the imagined situation verifies S, and that one has imagined that S. Overall, we can say that S is positively conceivable when one can imagine that S: that is, when one can imagine a situation that verifies S. (Chalmers, "Does conceivability entail possibility?", p. 150. [emphasis mine])So in cases of positive conceivability we must do two things. (1) First we must imagine a certain situation. (2) Second, we must determine that the situation verifies (or fails to verify) a given target proposition.
This two step process is characteristic of many of the most philosophically important "thought experiments". For example, the Gettier examples require us to first imagine a situation having certain characteristics and then determine whether or not a certain knowledge attribution is true in that situation. In this case, it is positively conceivable that one can have a justified true belief without knowledge (i.e., the situation doesn't verify the knowledge attribution). Hence, it is possible to have a justified true belief without knowledge. Hence, justified true belief isn't sufficient for knowledge.
Now the second part of Chalmer's definition relies essentially on the undefined notion of "verifying that a certain proposition is true" in a given situation or "having it be revealed that a certain proposition is true" in a given situation. And clearly, in cases of positive conceivability, we must be justified in believing that the imagined situation makes true (or fails to make true) the target proposition. Thus, positive conceivability leaves us with problem of justifying our belief that a given situation verifies (or fails to verify) the target proposition. Call this the problem of the alethic gap.
Notice that this problem doesn't arise for the intuitionist camp. For on that view, intuition is a form of basic evidence and it is just intuitive that the situation verifies the target proposition. But the conceivablists don't seem to be able to make a similar move. Are we supposed to find it conceivable that the situation verifies (or fails to verify) the target proposition? How would this differ from the original imaginative episode? And if it does differ, wouldn't we be off on a vicious regress?
If this is right, then the conceivability camp must rely on intuition (or something like it) in order to give a complete account of a priori evidence.