Order-dual relational semantics for non-distributive propositional logics

Abstract

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics (GGLs), introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with $n$ -ary additive or multiplicative operators ( $n$ -ary boxes and diamonds), with negation operators modally interpreted as impossibility and unnecessity (falsifiability), as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction (refutation) relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard (classical) interpretation (or dual interpretation) of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case.