arXiv subjects

Chrysafis Hartonas

Publications and source records attributed to Chrysafis Hartonas.

6 recordsLinked to original sources

arXiv2025

A General (Uniform) Relational Semantics for Sentential Logics

We present a general relational semantics framework which, by varying the axiomatization and components of the relational structures, provides a uniform semantics for sentential logics, classical and non-classical alike. The approach we take rests on a generalization of the J\'{o}nsson-Tarski representation (and duality) for Boolean algebras with operators to the cases of posets, semilattices, or bounded lattices (with, or without distribution) with quasi-operators. Completeness proofs rely on a choice-free construction of canonical extensions for the algebras in the quasivarieties of the equivalent algebraic semantics of the logics. Correspondence results for axiomatic extensions of the logics of implication that we study rely on a fully abstract translation into their modal companions and they are calculated using a generalized Sahlqvist - van Benthem algorithm.

cs.LO↗

arXiv2024

Distribution-Free Normal Modal Logics

This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, with a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn's Positive Modal Logic (PML). Unlike PML, we consider logics that may drop distribution and which are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of Intuitionistic Modal Logic.

cs.LO↗

arXiv2023

Choice-free Topological Duality for Implicative Lattices and Heyting Algebras

We develop a common semantic framework for the interpretation both of $\mathbf{IPC}$, the intuitionistic propositional calculus, and of logics weaker than $\mathbf{IPC}$ (substructural and subintuitionistic logics). This is done by proving a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free duality for the category of Heyting algebras and a category of topological sorted frames with a ternary sorted relation.

math.LO↗

arXiv2023

Choice-free Dualities for Lattice Expansions: Application to Logics with a Negation Operator

Constructive dualities have been recently proposed for some lattice based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining "choice-free spatial dualities for other classes of algebras [$\ldots$], giving rise to choice-free completeness proofs for non-classical logics''. We present in this article a way to complete the Holliday-Bezhanishvili project (uniformly, for any normal lattice expansion) by recasting recent relational representation and duality results in a choice-free manner. These results have some affinity with the Moshier and Jipsen duality for bounded lattices with quasi-operators, except for aiming at representing operators by relations, extending the J\'{o}nsson-Tarski approach for BAOs, and Dunn's follow up approach for distributive gaggles, to contexts where distribution may not be assumed. To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining canonical extensions in relational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases.

math.LO↗

arXiv2021

Duality for Normal Lattice Expansions and Sorted, Residuated Frames with Relations

We revisit the problem of Stone duality for lattices with various quasioperators, first studied in [14], presenting a fresh duality result. The new result is an improvement over that of [14] in two important respects. First, the axiomatization of frames in [14] was rather cumbersome and it is now simplified, partly by incorporating Gehrke's proposal [8] of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt's [11] recently proposed definition of bounded morphisms for polarities, though we need to strengthen the definition in order to get a Stone duality result. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be viewed as fragments of sorted, residuated (poly)modal logics, a research direction initiated in [16,17].

math.LO↗

arXiv2021

Reconciliation of Approaches to the Semantics of Logics without Distribution

This article contributes in that it clarifies and indeed completes an approach (initiated by Dunn and this author several years ago and again pursued by the present author over the last three years or so) to the relational semantics of logics that may lack distribution (Dunn's non-distributive gaggles). The approach uses sorted frames with an incidence relation on sorts (polarities), equipped with additional sorted relations, but, in the spirit of Occam's razor principle, it drops the extra assumptions made in the generalized Kripke frames approach, initiated by Gehrke, that the frames be separated and reduced (RS-frames). We show in this article that, despite rejecting the additional frame restrictions, all the main ideas and results of the RS-frames approach relating to the semantics of non-distributive logics are captured in this simpler framework. This contributes in unifying the research field, and, in an important sense, it complements and completes Dunn's gaggle theory project for the particular case of logics that may drop distribution.

math.LO↗