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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. [ABSTRACT FROM AUTHOR]

Following K. Segerberg [22] , D. Kozen [15] and V. Pratt [19] , who have been introduced dynamic propositional logic and dynamic algebras, dynamic propositional Łukasiewicz logic DP Ł (dynamic n -valued propositional Łukasiewicz logic D P Ł n) and dynamic MV -algebras (dynamic M V n -algebras) are introduced and theories of the logic DP Ł (D P Ł n) and dynamic MV -algebras (M V n -algebras) are developed. Dynamic MV -algebras (dynamic M V n -algebras) are algebraic counterparts of the logic DP Ł (D P Ł n), that in turn represent two-sorted algebras that combine the varieties of MV -algebras (M V n -algebras) (M , ⊕ , ⊙ , ∼ , 0 , 1) and regular algebras (R , ∪ , ; , ⁎) into a single finitely axiomatized variety (M , R , ◇) resembling R -module with "scalar" multiplication ◇. Kripke semantics is developed for dynamic propositional Łukasiewicz logic (dynamic n -valued propositional Łukasiewicz logic D P Ł n). [ABSTRACT FROM AUTHOR]

This paper introduces and studies extensions of modal logics by investigating the soundness of classical modal axioms in a weighted framework. It discusses the notion of relevant weight values, in a specific weighted Kripke semantics and exploits accessibility relation properties. Different generalisations of the classical axioms are constructed and, from these, a typology of weighted axioms is built, distinguishing between four types, depending on their relations to their classical counterparts and to the, possibly equivalent, frame conditions. [ABSTRACT FROM AUTHOR]