Pawel Pawlowski

In this paper, we contribute to the further development of non-deterministic semantics for propositional modality within the 8-valued framework. As our point of departure, we take MnD, a very weak modal logic where modal operators are inter-definable and uninterpreted. We study its extensions with respect to $\Box,\Diamond,\vee,\wedge,\rightarrow $ by means of simple refinements. We provide axiomatisations for all extensions obtained through these refinements. This systematic approach enables the exploration of a wide range of non-deterministic modal logics and hones our understanding of non-deterministic semantics for modality.

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 analyzed via non-deterministic semantics. Furthermore, other modal rules than the rule of necessitation were not yet characterized in the framework.In this paper, 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 interpretation 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.