This mini-workshop's purpose is to discuss recent developments and new ideas in modal logic.  This workshop is also held as PhilEth Seminar #214 of Laboratory of Philosophy and Ethics, Faculty of Humanities and Human Sciences, Hokkaido University. 

10th November 2023, 15:00-17:45

• 15:00--15:05 Opening

• 15:05--15:45 Katsuhiko Sano (Hokkaido University)

Analytic Cut and Mints' Symmetric Interpolation Method for Bi-intuitionistic Tense Logic

• 16:00--16:40 Yuki Nishimura (Tokyo Institute of Technology) 

Tableau Calculus for Undirected Graphs

• 17:00--17:40 Hitoshi Omori (Tohoku University)

Negation and modality in view of non-deterministic semantics

• 17:40--17:45 Closing

• Katsuhiko Sano (Hokkaido University): Analytic Cut and Mints - Symmetric Interpolation Method for Bi-intuitionistic Tense Logic

This talk establishes the Craig interpolation theorem of bi-intuitionistic stable tense logic BiSKt, which is proposed by Stell et al. (2016). First, we define a sequent calculus G(BiSKt) with the cut rule for the logic and establish semantically that applications of the cut rule can be restricted to analytic ones, i.e., applications such that the cut formula is a subformula of the conclusion of the cut rule. Second, we apply a symmetric interpolation method, originally proposed by Mints (2001) for multi-succedent calculus for intuitionistic logic, to obtain the Craig interpolation theorem of the calculus G(BiSKt). Our argument also provides a simplification of Kowalski and Ono (2017)'s argument for the Craig interpolation theorem of bi-intuitionistic logic. This is joint work with Hiroakira Ono (JAIST).

•  Yuki Nishimura (Tokyo Institute of Technology): Tableau Calculus for Undirected Graphs

An undirected graph is a kind of Kripke frame that has irreflexivity and symmetry. In this presentation, we formulate the tableau method of the hybrid logic for undirected graphs. Our main result is to show the completeness theorem and the termination property of the tableau method, which leads us to prove the decidability.  * This is a joint work with Tsubasa Takagi (TITech).

• Hitoshi Omori (Tohoku University):  Negation and modality in view of non-deterministic semantics

Non-deterministic semantics is a semantic framework systematically developed by Arnon Avron and Iddo Lev. One way of describing this semantics is that it generalizes a many-valued semantics. Generalization lies in the fact that the value of a complex formula is not determined uniquely by its subformulas, but instead there are several values that can be assigned. The aim of the talk is twofold. First we introduce the non-deterministic semantics without assuming any familiarity on the topic. Second, we discuss two related topics: (i) non-deterministic semantics for systems with very weak negation, and (ii) modal semantics *without* possible worlds.