All finitely axiomatizable subframe logics containing the provability logic CSM 0_{0} are decidable

In this paper we investigate those extensions of the bimodal provability logic CSM 0 (alias PRL 1 or F ?) which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all nitely axiomatizable subframe logics containing CSM 0 are decidable. We note that, as a rule, interesting systems in this class do not have the nite model property and are not even complete with respect to Kripke semantics.