☐ and ◇ in eight-valued non-deterministic semantics for modal logics

Abstract In this paper we study several extensions of the minimal modal logic M. This minimal modal logic is formulated in the language of classical propositional logic together with two modal operators and, which have no deductive power. By extending the Hilbert calculus for M with various axioms for and and/or the rule of necessitation, we obtain several well-known normal modal logics, as well as systems that are of pure theoretical interest. Those systems are shown to be sound and complete wrt to eight-valued semantics. Those semantics are obtained by refinements of an eight-valued semantics for M. Furthermore, we will briefly discuss some limitations of the method presented in this article.