The modal logic of probability

The present paper offers a novel axiomatization of the probability concept in terms of modal logic. The structures we axiomatize consist of a measurable space of possible worlds, and for each possible world, a probability measure on the space and a valuation function. Roughly speaking, these structures can be seen as probabilistic refinements of the familiar Kripke structures of modal logic. Leaving aside measurability restrictions, the difference between the two kinds of structures is simply that in a probability structure each world is mapped to a probability measure instead of a set. Conversely, a Kripke structure can be seen as an impoverishment of a probability structure, in which only supports (i.e., minimal closed sets of probability one) are considered. Similarly to Kripke structures, probability structures admit of various conceptual interpretations, but this paper was motivated by earlier work in epistemic logic and the foundations of decision theory and game theory, so we will be exclusively concerned here with the interpretation of probability as a measure of subjective belief.