Distinguished algebraic semantics for t -norm based fuzzy logics: Methods and algebraic equivalencies

Annals of Pure and Applied Logic 160 (1):53-81 (2009)
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Abstract

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit interval, the rational unit interval, the hyperreals , the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved

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Citations of this work

Structural Completeness in Fuzzy Logics.Petr Cintula & George Metcalfe - 2009 - Notre Dame Journal of Formal Logic 50 (2):153-182.
Fuzzy logic.Petr Hajek - 2008 - Stanford Encyclopedia of Philosophy.
Implicational logics III: completeness properties.Petr Cintula & Carles Noguera - 2018 - Archive for Mathematical Logic 57 (3-4):391-420.

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References found in this work

A propositional calculus with denumerable matrix.Michael Dummett - 1959 - Journal of Symbolic Logic 24 (2):97-106.
Logics without the contraction rule.Hiroakira Ono & Yuichi Komori - 1985 - Journal of Symbolic Logic 50 (1):169-201.
Fuzzy Sets.Lofti A. Zadeh - 1965 - Information and Control 8 (1):338--53.
Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.
A constructive analysis of RM.Arnon Avron - 1987 - Journal of Symbolic Logic 52 (4):939 - 951.

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