One question about subjective probability that I’ve received a few times is how necessarily false propositions can get a non-zero probability. It’s a consequence of any form of the axioms of probability that any tautologically false proposition must get probability zero, but depending on which form of the axioms one takes, this won’t have to be the case even for first-order logically false propositions, much less metaphysically impossible ones. The reason I say “depending on which form of the axioms one takes” is because there are several different forms that are generally considered to be equivalent.
For instance, Kolmogorov says that probability (1) is a non-negative function on sets of possible worlds (2) that assigns the value 1 to the complete set (3) and is (finitely or countably) additive on disjoint sets of worlds. Another formulation replaces (2) and (3), saying instead that logical truths get probability 1, and that probability adds on logically incompatible statements. Yet another (which I get from Brian Weatherson) replaces all four by saying that logical truths and falsehoods get probability 1 and 0 respectively, that if A entails B then P(A)≤P(B), and that P(A)+P(B)=P(A and B)+P(A or B).
The reason for this equivalence (and also, I think, the motivation behind the original questions) is that most people take a probability function (subjective, epistemic, logical, objective, or whatever) to be a function from sets of possible worlds to real numbers. (I’ve also heard of some proposals that would restrict to rationals, expand to complex numbers, or change to p-adics or some other number system. However, I think most of these suggestions occur for purely mathematical reasons and just encode certain similarities to probabilities, but aren’t really considered to be probabilities themselves.) The later formulations I give above talk about logical truths and the like, but it is very easy (and relatively standard) to talk about propositions as just being sets of worlds. And if the set of all possible worlds is taken to include all logically possible worlds (rather than metaphysically possible worlds), then I believe all the above formulations are equivalent. This slip between logical and metaphysical possibility is I think why people worry about whether impossible statements must get probability 0.
However, in many cases it seems natural to take a more neutral stance (as Brian Weatherson did) and just think of a probability function as an assignment of real numbers to propositions or even just statements in a language rather than sets of possible worlds. This was pursued by Karl Popper in appendices *iv and *v to The Logic of Scientific Discovery, where he tries to give an account of conditional probability on antecedents of probability zero. (I happen to think his particular approach is wrong, but he has the right general idea.) Popper’s axioms are quite nice, in that he can prove for any two tautologically equivalent statements that they get the same probability from the probability axioms alone, without making any assumptions about what the logical connectives mean or what propositions are.
I think that for dealing with tricky logics, or statements containing essential indexicals, this sort of distinction is important to make. People try to modify the “sets of possible worlds” approach by talking about “sets of centered possible worlds”, but I’m skeptical about how much can be accomplished in this way.