Another extension of Basic Logic

Basic propositional logic was first introduced by Albert Visser in [6]. Basic logic is a proper subsystem of intuitionistic logic, and can be constructed from it by a weakening of modus ponens. This weakening allows for a larger collection of theories, for example formal provability logic. Formal provability logic is related to the provability modal logic GL just as intuitionistic logic is related to modal logic S4. Basic logic itself is related to K4. Another motivation for the study of Basic Logic is based on a search for a new ‘constructive’ interpretation of the connectives, see [4] or [5]. Basic logic may be characterized as the logic of the generalized Kripke models in which the accessibility relation need not be reflexive. Different aspects of Basic logic is investigated in [2], see also [1]. In this talk we present an extension of Basic logic that can be characterized as the logic of the generalized Kripke models with motive: if there is an end of the worlds, it should be reflexive.