ИСТИНА

We give a sufficient condition for Kripke completeness of modal logics that have the transitive closure modality. More precisely, we show that if a modal logic admits what we call definable filtration, then its enrichment with the transitive closure modality (and the corresponding axioms) is Kripke complete; in addition, the resulting logic has the finite model property and admits definable filtration, too. This argument can be iterated, and as an application we obtain the finite model property for PDL-like expansions of multimodal logics that admit definable filtration. At the end of the talk we formulate problems that are left open.