Modal and temporal extensions of non-distributive propositional logics
Chrysafis Hartonas · Logic Journal of IGPL · 2015
A notorious difficulty with modal extensions over a non-distributive propositional basis is to construct canonical Kripke models (time flow structures, when a temporal interpretation is intended) that respect the well-established intuitions about the meaning of modal (temporal) operators. Indeed, advances in modal logics over a non-distributive propositional basis over the last decade or so address a number of issues of significance, such as Sahlqvist (algorithmic) correspondence and completeness, yet they do this while resting on a notion of frame and model, canonical or otherwise, that compromises the intuitive semantics of the modal operators in important ways. This becomes particularly apparent when a dynamic, or a temporal reading of boxes and diamonds is intended. This article is restricted to the simplest temporal logic, a Priorean Tense Logic over a negation and implication free non-distributive propositional basis. We build up to this system by considering separately systems with modal operators in isolation, or in related groups, and we prove, for each system, completeness via a traditional canonicity argument. We establish that the absence of a distributivity assumption of conjunctions over disjunctions and conversely has no effect on the interpretation of boxes and diamonds, which are interpreted exactly as in classical normal modal logics. The motivation of this work resides in the realization that at least part of the reason for the manifest dissatisfaction with studying and applying modal logics over a non-distributive propositional basis is due precisely to the fact that a large number of semantic intuitions, results and techniques familiar from distributive logics seem to have to be inescapably abandoned. We argue in this article that this is not necessarily the case. The results we present can be applied, e.g. in studying dynamic, or temporal extensions of Orthomodular Quantum Logic, as well as in the relational semantics for Substructural Logics.