Group in Logic and the Methodology of Science

This paper is about the relation between two kinds of models for propositional modal logic: possibility models in the style of Humberstone and possible world models in the style of Kripke. We show that every countable possibility model M is completed by a Kripke model K, its worldization; every total world of K is the limit of more and more refined possibilities in M, and every possibility in M is realized by some total world of K. In addition, we define a general notion of a possibilization of a Kripke model, which is a possibility model whose possibilities are sets of worlds from the Kripke model. We then characterize the class of possibility models that are isomorphic to the possibilization of some Kripke model. In particular, every possibility model in this class can be represented as a possibilization of one of its worldizations; and every possibility model can be naturally transformed into one in this class by, for example, deleting duplicated possibilities. This representation theorem clarifies the relationship between possibility models and Kripke models.