I think the “logical probability” problem, of “how do I quantify my beliefs about the trillionth digit of \pi” and whatnot, is probably just an issue of domain-theoretic maximum-entropy distributions: each step of computation can give us more information that can be used to concentrate the measure better, and domain theory says how computational results are built out of other results from computations that may not have finished yet.
I think the “logical probability” problem, of “how do I quantify my beliefs about the trillionth digit of \pi” and whatnot, is probably just an issue of domain-theoretic maximum-entropy distributions: each step of computation can give us more information that can be used to concentrate the measure better, and domain theory says how computational results are built out of other results from computations that may not have finished yet.