In previous publications, it was shown that finite non-deterministic matrices are quite powerful in providing semantics for a large class of normal and non-normal modal logics. However, some modal logics, such as those whose axiom systems contained the Löb axiom or the McKinsey formula, were not be analyzed via non-deterministic semantics. Furthermore, other modal rules than the rule of necessitation were not yet characterized in the framework.

In this presentation, we will overcome this shortcoming and present a novel approach for constructing semantics for normal and non-normal modal logics that is based on restricted non-deterministic matrices. This approach not only offers a uniform semantical framework for modal logics, while keeping the meaning of the involved modal operators the same, and thus making different systems of modal logic comparable. It might also lead to a new understanding of the concept of modality.