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).
