David picks up on my idea of contextual impossibility, and makes the interesting suggestion that we can believe indirect paradoxes, but no-one can really believe a direct paradox (e.g. "this sentence is false"). Indirect paradoxes are different, because we might not realize that the paradoxical context obtains, and thus we might believe the claim to be simply true or false.
As David writes:
Imagine a page on which three statements are written:
(a) Most statements on this page are false.
(b) Paris is the capital of France.
Suppose the person who writes up this page mistakenly believes that Paris is the capital of Italy, not France. This person believes that (b) and (c) are false, and therefore that (a) is true...
Consider the proposition "This sentence is false." I might mistakenly believe I believe this proposition, but there is no proposition there to believe; the sentence is a mere combination of words. But... in the above example, I can believe "Most sentences on this page are false" as long as I believe that Paris is not the capital of France.
But we should also say that (a) is, in this context, a "mere combination of words", for if it expressed a genuine statement then we would have the paradoxical result that it is true iff it is false. Note that I do not deny that the person can believe the proposition most statements on the page are false. Rather, I deny that (a) expresses this proposition. The proposition itself is actually false, since there are only two statements on the page - (b) and (c) - and exactly half, not "most", of them are false.
So the paradox is stuck on the page, and doesn't carry over into our beliefs. Could we come up with a belief-variant of this paradox? Sure. Suppose that I have only the following three beliefs:
(a') Most of my beliefs are mistaken.
(b') Paris is the capital of France.
My belief (a') is mistaken iff it is not. Contradiction. So we should deny that (a') was a real belief, i.e. deny that it genuinely expressed a proposition. In the given context, I cannot really believe (a') after all. I might mistakenly believe that I believe it, but "there is no proposition there to believe; the sentence is a mere combination of words."
So David is mistaken that we can believe indirect/contextual paradoxes. In many cases we can believe a proposition that would be paradoxical if expressed in the appropriate context. But this is only because our beliefs provide a new 'context', in which the proposition is no longer paradoxical. If we construct an indirect paradox that carries over into our thoughts, then it is impossible to have the paradoxical belief. In this respect, direct and indirect paradoxes behave the same way after all.