A non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts

TL;DR

This paper introduces a non-distributive hypersequent calculus for semiconcepts of contexts and extends it with modal logic based on Kripke contexts, providing semantics, soundness, completeness, and applications to conceptual knowledge and rough set theory.

Contribution

It develops a novel non-distributive hypersequent calculus and modal extension with Kripke semantics for semiconcepts, linking logic, algebra, and rough set theory.

Findings

Soundness and completeness of the systems with respect to relational semantics.

Modal extensions capture properties of various Kripke contexts.

Framework expresses conceptual knowledge and approximations in rough set theory.

Abstract

A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its modal extension \textbf{MPDBL} are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for \textbf{PDBL} is next proposed, where any formula is interpreted as a semiconcept of a context. For \textbf{MPDBL}, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding appropriate sequents to \textbf{MPDBL} results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. One of these systems is a logic for topological pure double Boolean algebras. It is demonstrated that, using \textbf{PDBL}, the basic notions and relations of conceptual knowledge can be expressed and inferences involving negations can be obtained. Further, drawing a connection with rough set theory, lower and upper approximations of semiconcepts of a context are defined. It is then shown that, using the formulae and sequents involving modal operators in \textbf{MPDBL}, these approximation operators and their properties can be captured.