If you learned that the god of [insert religion] is real, would all bets be off?

[Belacqua]

I guess where you and I disagree is that I believe you're being too charitable when you say "MAYBE everything I know about logic is simply a local and contingent misunderstanding. Since I don't have the knowledge that (we imagine) God would have, I have to admit I might be entirely wrong"

I, on the other hand, firmly believe there is zero chance that everything I think I know about logic can turn out to be a misunderstanding. Some of the more complicated things in logic, maybe, but not everything, because that would include things like the law of identity and the law of the excluded middle.

If a god tried to trick me into thinking he/she/it has produced a married bachelor, I would confidently hold to my belief that this is impossible.

This is where my concern with whether I'm being dogmatic comes in. Is my unwillingness to let go of logic a type of dogmatism?

OK, we're quickly getting over my head here. (Which I admit isn't hard to do.)

I have vaguely heard of things like alternative mathematical systems, in which the rules of mathematical logic that we learned in school don't apply. Non-Euclidean, something like that?

Is it possible that aliens could have a different logic than we do? 

I don't see how they could -- I agree with you that the law of identity seems immutable to me.