The CEMS.UL - Center for Mathematical Studies is promoting the seminar "Higher-order Kripke models for intuitionistic and other non-classical modal logics", with the participation of Victor Barroso-Nascimento (University College London)
Abstract: In this talk I will present higher-order ("nested") Kripke models, a generalization of Kripke models that is remarkably close to Kripke's original idea -- both mathematically and conceptually. Intuitionistic modal logics are used for the case study, as the generalisation is arrived at specifically after the paradigmatic cases of intuitionistic modal logics IK and MK are studied. Standard models are now 0-ary models, whereas n-ary models for n > 0 are models whose set of objects (''possible worlds'') contain only (n-1)-ary models. A key idea is the use of worlds as fixed points for modal definitions, in the sense that what is necessary or possible in a world of a frame depends only on what is true in the same world on the accessible frames. The association between conditions on accessibility relations and modal axioms also carries over to this framework, so modal logics stronger than K can be obtained by imposing requirements on the relations between frames. Just like Kripke models define a concept of ''alternative'' for classical models, the n-ary models (for n > 0) defines the same concept for any interpretation of the (n-1)-ary models. In particular, if we use the common interpretation of intuitionistic Kripke models as representing the activities of a mathematician in a particular timeline, modalities are defined in a natural way through the concept of alternative timelines.