Non-Deterministic Many-Valued Modal Logic

In this paper, we extend the techniques of non-deterministic semantics for propositional logics introduced by Arnon Avron and Iddo Lev to provide an account of modal operators. We examine the counterparts of the standard concepts from deterministic modal logics in this setting, giving parallel accounts of necessity, possibility, frame conditions, and so forth. As a concrete application, we examine the logic C
min
, introduced by Walter Carnielli and João Marcos as a variant of da Costa's C-systems. We consider how it can be augmented with a necessity operator and detail analogies to stronger normal systems. Finally, we examine an extension of classical two-valued logic with a non-deterministic modality and discuss some of its properties and applications.