Many-Valued Logics

Preprint submitted to Elsevier Science. 8 May 2005. — 56 p.[Siegfried Gottwald: Institute of Logic and Philosophy of Science, Leipzig University, Leipzig, Germany].The paper considers the fundamental notions of many-valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics.Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics.Basic ideas.From classical to many-valued logic.Particular truth degree sets.Designated truth degrees.Logical validity and logical consequence.Outline of the history.Basic Systems of Many-Valued Logics.The G¨odel logics.The Lukasiewicz logics.The Product logic.The Post logics.Standard and Algebraic Semantics.Boolean algebras.Godel and Lukasiewicz logics.Product logic.Post logics.Particular Three- and Four-Valued Systems.Three-Valued Systems.Four-Valued Systems.Logics with T-Norm Based Connectives.Residuated Implications versus S-Implications.Continuous T-Norms.The Logic of Continuous T-Norms.The Logic of Left Continuous T-Norms.Some Generalizations.Pavelka Style Extensions.Gerla‘s General Approach.Some Recent Applications.Fuzzy sets theory.Non-monotonic fuzzy reasoning.References (150 publ).