This will be a joint session with the LLAMA seminar, with two speakers.
Date and time: Wednesday, May 7th 2025, 15:00-18:00
Venue: ILLC Seminar Room F1.15, Science Park 107 and online.
First talk (starting at 15:00)
Speaker: Katsuhiko Sano (Hokkaido University)
Title: Craig Interpolation for Bi-intuitionistic Stable Tense Logic
Abstract. Bi-intuitionistic stable tense logic (BiSKt), introduced by Stell et al. (2016), provides a logical framework for mathematical morphology on graphs consisting of nodes and edges. In this talk, we establish the Craig interpolation theorem for BiSKt from a proof-theoretic perspective — that is, in terms of a sequent calculus for BiSKt. Although our sequent calculus is not cut-free, applications of the cut rule can be restricted to analytic ones — namely, those in which the cut formula is a subformula of the conclusion of the cut rule. To establish the Craig interpolation theorem for this calculus, we use a symmetric interpolation method originally proposed by Mints (2001) for the multi-succedent calculus of first-order intuitionistic logic. This method can be seen as a generalization of Maehara’s method. Our proof-theoretic approach also simplifies the method developed by Kowalski and Ono (2017) for proving the Craig interpolation theorem in bi-intuitionistic logic. This is a joint work with Hiroakira Ono (JAIST).
Second talk (starting at 16:30)
Speaker: Alexander Kurz (Chapman University)
Title: Quantale-Valued Modal Logic
Abstract. For applications of logic it is often desirable to have a common umbrella encompassing both classical discrete two-valued and quantitative continuous many-valued reasoning. What are principled ways to extend 2-valued modal logic to many-valued modal logic? What is a suitable generalization of Kripke semantics to this setting?
Without assuming knowledge of category theory, we will explain how category theory allows us to build a general framework to answer these questions. The key observation is that 2 is not only the familiar Boolean algebra but also an instance of a more general structure known as a quantale. A quantale is simply a complete lattice with a little extra structure. As observed by Lawvere in 1973, this extra structure allows us to extend ordinary 2-valued logic to a generalized logic that takes truth values in an arbitrary quantale. The models of such generalized logics encompass a variety of structures including metric spaces. In this talk, we will give a general introduction to this area of logic and also sketch some novel results about canonical extensions of fuzzy-algebras obtained in collaboration with Apostolos Tzimoulis.