Modal logic, fundamentally

Abstract

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices $L$ equipped with an antitone operation $\neg$ sending $1$ to $0$, a completely multiplicative operation $\square$, and a completely additive operation $◊$. Such lattice expansions can be represented by means of a set $X$ together with binary relations $⊲$, $R$, and $Q$, satisfying some first-order conditions, used to represent $(L,\neg )$, $\square$, and $◊$, respectively. Indeed, any lattice $L$ equipped with such a $\neg$, a multiplicative $\square$, and an additive $◊$ embeds into the lattice of propositions of a frame $(X,⊲,R,Q)$. Building on our recent study of "fundamental logic", we focus on the case where $\neg$ is dually self-adjoint ($a\le \neg b$ implies $b\le \neg a$) and $◊\neg a\le \neg \square a$. In this case, the representations can be constrained so that $R=Q$, i.e., we need only add a single relation to $(X,⊲)$ to represent both $\square$ and $◊$. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures $(X,⊲,R)$.

Forthcoming in Advances in Modal Logic, Vol. 15