In this paper we investigate those extensions of the bimodal provability logic C⃗ SM0 (alias P⃗ RL1 or F⃗ −) 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 finitely axiomatizable subframe logics containing C⃗ SM0 are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics.

The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both finite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.

This chapter is a continuation of the preceding one, and we begin it at the place where the authors of Basic Modal Logic left us about fifteen years ago. Concluding his historical overview, Krister Segerberg wrote: “Where we stand today is difficult to say. Is the picture beginning to break up, or is it just the contemporary observer’s perennial problem of putting his own time into perspective?” So, where did modal logic of the 1970s stand? Where does it stand now? Modal logicians working in philosophy, computer science, artificial intelligence, linguistics or some other fields would probably give different answers to these questions. Our interpretation of the history of modal logic and view on its future is based upon understanding it as part of mathematical logic.