This should be really obvious, but I can’t find any convenient link, and I can’t see how to do this myself – at least not easily.

Say you wanted to mix probability theory and modal logic in the following way. The language of the sentences over which the probability function is defined includes a box, here written L. Whenever p is a sentence, Lp is a sentence, so all the usual formation rules apply. The logic for the box is S4. (We can substitute in other logics later once we figure out how S4 works.) We want the probability functions to be all and only the measure functions on Kripke models for S4. So Pr is an S4-probability function iff there is a Kripke model [W, R, V] and a measure function m defined over W such that for any sentence A, Pr(A) = m({w: A is true in [W, R, V] at w}). (Note I’m using square brackets around W, R, V because angle brackets confuse the HTML coder.)

Anyway, say you wanted to do all of that. How would you go about drawing up axioms to characterise the class of probability functions so isolated? Here’s one hypothesis. You’d simply take the axioms for classical probability theory, which always include (often tacitly, but always) some reference to either an entailment function or a class of logical truths. You then interpret that reference as a reference to S4-entailment, or S4-logical truth. And there’s your axiom system.

Does it work? I’ve got no idea. One of the things I proved in my paper on “intuitionist probability logic”:http://brian.weatherson.org/conprob.pdf is that this approach really won’t work when you want to move from classical logic to intuitionist logic. The problem is that classically equivalent axiomatisations of the probability calculus turn out to be inequivalent when ‘re-interpreted’ in an intuitionist way. (I.e. the references to entailment or logical truth are taken to be references to _intuitionist_ entailment or _intuitionist_ logical truth.) Could the same thing happen for the move from classical logic to S4? I don’t know. I thought it might be possible to generalise by claim about intuitionist logic to show that the answer to this question was _no_, but on a little reflection I rather doubt this is true. I think this is just an open question.

A (very brief!) lit search reveals very little on the intersection between probability theory and modal logic. Williamson’s 1998 BJPS paper has some relevant material, but it’s not exactly on point. The probabilistic semantics literature also has interesting material (esp by Cross and by Morgan) but it doesn’t seem (from what I can find from here late at night) to be exactly relevant either. (In all cases this lack of relevance is not due to shortcomings of the authors, but just because they were asking and answering different questions. I can sometimes appear overly critical here, so I should be clear that _in this case_ no criticism is intended!) But probability and modality are such big topics one would think there’d be _something_ on their intersection, and I don’t think the way I’ve framed the problem is _entirely_ idiosyncratic.