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京都大学数理解析研究所の佐藤です。
10月29日11:00から、チェコ科学アカデミーのPetr Cintula氏に 以下の講演をしていただくことになりましたので、 ご連絡いたします。どうぞお気軽にお越しください。 ========== Time: 11:00-12:00, 29 Oct, 2015 Place: Rm 478, Research Building 2, Main Campus, Kyoto University 京都大学 本部構内 総合研究2号館 4階478号室 http://www.kyoto-u.ac.jp/en/access/yoshida/main.html (Building 34) http://www.kyoto-u.ac.jp/ja/access/campus/map6r_y.htm (34番の建物)
Speaker: Petr Cintula (The Czech Academy of Sciences)
Title: Logic and mathematics with lattice-valued predicates
Abstract: Classical predicate logic interprets n-ary predicates as mappings from the n-th power of a given domain into the two-valued boolean algebra 2. The idea of replacing 2 by a more general structure is very natural and was shown to lead to a very interesting mathematics: prime examples are the boolean-valued or Heyting-valued models of set theory (or even more general models proposed by Takeuti & Titani (1992), Titani (1999), and Hajek & Hanikova (2001)).
In the first part of the talk we present a framework for the study of logics where predicates can take values in a lattice (with additional operators) from a given class satisfying certain minimal conditions (our framework covers previous approaches of Rasiowa & Sikorski (1963), Horn (1969), Rasiowa (1974), Hajek (1998), and others). For each such logic we first describe its `propositional' part and then use it to give an axiomatization of the full first-order logic.
The second part of the talk shows that the proposed logical formalism is rich enough to support non-trivial mathematical theories. We illustrate it by proving `graded' variant of the well-know relation between equivalences and partitions. The goal of the example is to illustrate the contrast between very general semantical interpretation of the proven fact and its almost classical proof.