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

Abstract

A non-distributive two-sorted hypersequent calculus PDBL and its modal extension MPDBL are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for PDBL is next proposed, where any formula is interpreted as a semiconcept of a context. For 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 MPDBL results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. In particular, one gets the logics MPDBL4 and MPDBL5 with semantics respectively based on reflexive, transitive Kripke contexts and reflexive, symmetric and transitive Kripke contexts. MPDBL4 also happens to be the logic for topological pure double Boolean algebra. It is demonstrated that, using 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 MPDBL5, these approximation operators and their properties can be captured.