Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system (...) and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)

In [12] the study of Positive Modal Logic is initiated using standard Kripke semantics and the positive modal algebras are introduced. The minimum system of Positive Modal Logic is the -fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed (...) in [12]. The frames of this semantics consists of a set of indexes, a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasi-order of the frame. This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqist Theorem for sequents and the semantics proposed in [6]. Also a Goldblatt-Thomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved. (shrink)

This first textbook on formal concept analysis gives a systematic presentation of the mathematical foundations and their relations to applications in computer science, especially in data analysis and knowledge processing. Above all, it presents graphical methods for representing conceptual systems that have proved themselves in communicating knowledge. The mathematical foundations are treated thoroughly and are illuminated by means of numerous examples, making the basic theory readily accessible in compact form.

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.