Modal Logic Over Finite Structures

In this paper, we develop various aspects of the finite model theory of propositional modal logic. In particular, we show that certain results about the expressive power of modal logic over the class of all structures, due to van Benthem and his collaborators, remain true over the class of finite structures. We establish that a first-order definable class of finite models is closed under bisimulations if it is definable by a `modal first-order sentence’. We show that a class of finite models that is defined by a modal sentence is closed under extensions if it is defined by a diamond-modal sentence. In sharp contrast, it is well known that many classical results for first-order logic, including various preservation theorems, fail for the class of finite models.