Consequence and degrees of truth in many-valued logic

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Josep Maria Font
Title of the chapter: Consequence and degrees of truth in many-valued logic
Title of the book: Petr Hájek on Mathematical Fuzzy Logic

Franco Montagna
Series: Outstanding Contributions to Logic
Volume: 6
Pages: 117–142
Publisher: Springer-Verlag
City: Heidelberg-Berlin-New York
Year: 2015


I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of Bolzano’s idea of consequence as truth-preserving when “truth comes in degrees”, as is often said in many-valued contexts, than the usual scheme that preserves only one truth value. I review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic in the framework of Łukasiewicz logics and in the broader framework of sub- structural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices. I also review some scattered, early results, which have appeared since the 1970’s, and make some proposals for further research.