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alex_zag_al comments on Applications of logical uncertainty - Less Wrong
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If you automatically assign the axioms an actually unobtainable certainty, you don't get the rational degree of belief in every proposition, as the set of "propositions" includes those not conditioned on the axioms.
Hmm. Yeah, that's tough. What do you use to calculate probabilities of the principles of logic you use to calculate probabilities?
Although, it seems to me that a bigger problem than the circularity is that I don't know what kinds of things are evidence for principles of logic. At least for the probabilities of, say, mathematical statements, conditional on the principles of logic we use to reason about them, we have some idea. Many consequences of a generalization being true are evidence for a generalization, for example. A proof of an analogous theorem is evidence for a theorem. So I can see that the kinds of things that are evidence for mathematical statements are other mathematical statements.
I don't have nearly as clear a picture of what kinds of things lead us to accept principles of logic, and what kind of statements they are. Whether they're empirical observations, principles of logic themselves, or what.