Filtration is one of the main methods for establishing the finite model property (fmp) in modal logic. Stable modal logics are normal modal logics that are well-behaved with respect to this method, so, in particular, all stable modal logics have the fmp. The class of stable modal logics was recently introduced as a class of logics that are axiomatizable by some special stable canonical rules. In this talk, we will further study the class of stable modal logics. First, we will give a semantic characterization of stable modal logics over the basic modal logic K. Then, by using transitive filtrations, we will adjust the notion of stability to normal modal logics over K4 for which we obtain an even more desirable characterization result. Finally, we will show that many standard modal logics fall into our class and that there is a continuum of stable modal logics over K and also over K4.

This is joint work with G. and N. Bezhanishvili.