Advances in Nonstandard Logic

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Main aim of this project is to promote the study of nonstandard logic and its application. When starting the project, we have chosen the following four as major subjects of our study :1. algebraic approach to substructural logics,2. epistemic logic and game theory,3. study of nonstandard predicate logics,4. proof-theoretic approach to nonstandard logics.Here are major achievments of our project :1. In collaboration with T.Kowalski, H.Ono has been developing a study of varieties of residuated lattices as algebraic semantics for logics without contraction rule. He has completed four joint papers, two of which have already been published in international journals. At the same time, H.Ono has given four invited talks on this topics at Torun, Poland, 1998, Barcelona, Spain, 1999, Milano, Italy, 1999, and Mantova, Italy, 2000. Recently, Kowalski and Ono have written up a monograph "Residuated Lattices".2. Kaneko and Suzuki have been developing a comprehensive study of game logic and common k … More nowledge logic as logical frameworks of game theory. Kripke-completeness and incompleteness problems of minimum extensions of modal predicate logics are discussed by Shimura and One, while Suzuki introduced stronger semantics for nonstandard predicate logics. On sequent calculi and natural deduction systems in relation to lambda calculi, Komori, Hirokawa, Takano and Kashima obtained interesting results. All of these results have been either published in international journals or announced at international conferences.3. As a part of the activity of our project, we organized two small international workshops. In 1999, a workshop "Algebra ＆ Substructural Logics" was held, where eight people were participated from abroad as invited speakers. Selected papers of the workshop have been published in a special issue of Reports on Mathematical Logic in 2001. In 2000, we organized a workshop "Algebra and Logic" by inviting three people from abroad as guest speakers.4. We supported 32nd, 33rd and 34th Annual Meetings of Mathematical Logic. The Meeting is regarded as the most important meeting on nonstandard logic in Japan, to which members of our project have contributed as active speakers for many years. Proceedings of these three meetings were published and distributed widely. Also, selected papers of the 34th Annual Meeting will be published as a special issue of Bulletin of the Section Logic in 2001. Less