counterfactuals, probability, and logic

There's a natural connection between set theory and logic that can be more or less drawn by considering the set of possible universes, and making a correspondence between a binary statement ("watermelon is a fruit") and the set of universes in which it's true.  The statement "A and B" is true in exactly those universes in the intersection of the two sets; logical "and" is equivalent to set intersection.  "or" is the union.  "not" is the complement relative to the set of possible universes.  "A implies B" means "either A is false or B is true".[1]

We can extend this common notion of logic and sets by introducing probability theory.  For any probability distribution on the set of universes, there's a probability that A is true, and a probability that B is true.  If we know (for all elements of the set) that A implies B, then we know that for any probability distribution, the probability that A is true is less than or equal to the probability that B is true; perhaps less obviously, the converse is also true, at least for finite sets of universes: if it is the case that the probability that A is true is less than or equal to the probability that B is true no matter what valid probability distribution is used, then A implies B.  If we restrict to one probability distribution, or a proper subset of all possible probability distributions, then there might be more to say; in particular, with one distribution, we can do Bayesian inference, and since P(A|A)=1, we have that if A implies B, P(B|A)=1.

Suppose we ask, "what would have happened if the Axis had won World War II?"  To some extent the answer necessarily depends on how we fill out the counterfactual.  In settings where we feel as though we have a reasonable answer to a counterfactual question like this, I think it's because we think there is some distribution (or distributions) of universes that is somehow "reasonable", and that, conditional on the information provided in the counterfactual, that answer is more likely than its complement.  For questions that are particularly ill-formed[2] may suffer from being conditional on far-fetched possibilities, but also may suffer from conditioning on information that is relatively independent of other interesting information; A is interesting about B if P(B|A) is close to 0 or 1 and substantially different from P(B).

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[1] You might ultimately know which universe you're in, or that you're in one of a restricted set of universes, but that's separate from the concerns of formal logic.

[2] I have in mind, in particular, the sort of question my son asks, e.g. "What if a baby beat a grandmaster in chess?", which tend to have the additional flaw that it's not clear what about the proposed reality is being asked.