  We give a sufficient condition for Kripke completeness of modal logics
enriched with the transitive closure modality. More precisely, we show that if
a logic admits what we call definable filtration (ADF), then such an expansion
of the logic is complete; in addition, has the finite model property, and again
ADF. This argument can be iterated, and as an application we obtain the finite
model property for PDL-like expansions of logics that ADF.

Logic