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Introduction to mathematical fuzzy logic Petr Cintula Ústav informatiky AV CR1 Carles Noguera ZS 2013/2014 čas přednášky bude stanoven emailovou úmluvou2 Originating as an attempt to provide solid logical foundations for fuzzy set theory, and motivated also by philosophical and computational problems of vagueness and imprecision, mathematical fuzzy logic (MFL) has become a significant subfield of mathematical logic. Research in this area focuses on many-valued logics with linearly ordered truth theories and challenging problems, thus continuing to attract an ever increasing number of researchers. The goal of this course is to provide an up-to-date introduction to MFL. Starting with the motivations and historical origins of the area, we present MFL, its main methods, and its core agenda. In particular, we focus on some of its better known logic systems (Lukasiewicz and Gödel–Dummett logics, HL, MTL) and present a general theory of fuzzy logics. Finally, we give an overview of several currently active lines of research in the development and application of fuzzy logics. 1. Motivations and historical origins of mathematical fuzzy logic. Basic notions of Algebraic Logic. Introduction to Lukasiewicz and Gödel–Dummett propositional logics: proof systems and their algebraic counterparts. 2. Main properties of Lukasiewicz and Gödel–Dummett logics: deduction theorem, completeness with respect to chains, standard and rational completeness, lattice of axiomatic extensions, game-theoretic semantics, finite model property, computational complexity, and functional representation. 3. Extending Lukasiewicz and Gödel–Dummett logics: richer propositional languages, modalities, first-order formalisms. Undecidability, axiomatizability, and arithmetical hierarchy. 4. T-norm based fuzzy logics HL and MTL. The growing family of fuzzy logics. Weakly implicative semilinear logics as a general theory for fuzzy logics. 5. Applications and further lines of research in fuzzy logics: first-order and higher systems, logics for uncertain events, modalities, paraconsistent reasoning, description logics, and counterfactuals. Study literature [1] Petr Cintula and Petr Hájek and Carles Noguera (eds.). Handbook of Mathematical Fuzzy Logic, Studies in Logic, Mathematical Logic and Foundations, vol. 37 and 38, College Publications, London, 2011. [2] Siegfried Gottwald. A Treatise on Many-Valued Logics, volume 9 of Studies in Logic and Computation. Research Studies Press, Baldock, 2001. [3] Petr Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998. 1 2 Pod Vodárenskou věžı́ 2, poblı́ž stanice metra Ládvı́, www.cs.cas.cz. Zájemci o účast mohou poslat své časové možnosti do 29.9.2013 na email [email protected]