Preface

The 33rd MLG meeting was held at Echigo-Yuzawa, Japan on January 10– 13, 2000. The meeting was supported mainly by Grant-in-Aid for Scientific Research (B)(1) No. 10440027, Advances in Nonclassical Logic and for Exploratory Research No. 11874016, Algebra for Substructural Logics, both of whose head investigator is Hiroakira Ono, Japan Advanced Institute of Science and Technology, Hokuriku.

There were 24 participants. This volume contains extended abstracts of the talks presented at the meeting.

We are extremely grateful to all the speakers and participants who made this meeting successful. We hope this volume will contribute to the further progress of mathematical logic and its related area.

Yuichi KOMORI Takafumi SAKURAI January 24, 2000

Talbe of Contents

A modified subformula property for the modal logic K5 1 Mitio Takano

A Natural Deduction for Classical Logic,

in which there exists a proof with the Subformula Property 3 Yuichi Komori

Labelled Sequent Calculi and Completeness Theorems

for Implicational Relevant Logics 7 Ryo Kashima

Semantics and Cut-free Provability of Modal Substructural Logics 11 Takafumi Sakurai

Similarity on a tableau method between modal propositional logic S5

and classical propositional logic Cl 15 Yasuhito Suzuki

Products of Modal Logics by General Frames 19 Yasusi Hasimoto

On a modal predicate logic with Glivenko type characterization 23 Tatsuya Shimura

Quantum set theory 27 Satoko Titani

A real vector space which has a base

but which has no well-orderable one 31 Nobutaka Tsukada

Epistemic Models of Shallow Depths and Decision Making in Games 34 Mamoru Kaneko and Nobu-Yuki Suzuki

A new system of common knowledge predicate logic 38 Takashi Nagashima and Yoshihito Tanaka

Ultraproduct of Kripke sheaves and Halldén-completeness

in super-intuitionistic predicate logics 42 Nobu-Yuki Suzuki

Propositional Dynamic Logic has interpolation 46 Tomasz Kowalski On interpretability logics 49

Katsumi Sasaki

Cut-free Sequent Calculi for Visser’s Propositional Logics 53 Kentaro Kikuchi

Logics of finite width for Some Intuitionistic Modal Logic 57 Kazuyo Inoi