Logic Journal of the IGPL 25 (2):145-182 (2017)

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, 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, with negation operators modally interpreted as impossibility and unnecessity, 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 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 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.
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DOI 10.1093/jigpal/jzw057
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References found in this work BETA

Algebraic Methods in Philosophical Logic.J. Michael Dunn - 2001 - Oxford, England: Oxford University Press.
Semantic Analysis of Tense Logics.S. K. Thomason - 1972 - Journal of Symbolic Logic 37 (1):150-158.
Star and Perp: Two Treatments of Negation.J. Michael Dunn - 1993 - Philosophical Perspectives 7:331-357.
Positive Modal Logic.J. Michael Dunn - 1995 - Studia Logica 55 (2):301 - 317.

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Citations of this work BETA

Stone Duality for Lattice Expansions.Chrysafis Hartonas - 2018 - Logic Journal of the IGPL 26 (5):475-504.
Lattice Logic as a Fragment of (2-Sorted) Residuated Modal Logic.Chrysafis Hartonas - 2019 - Journal of Applied Non-Classical Logics 29 (2):152-170.

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