Distinguished algebraic semantics

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Petr Cintula
Francesc Esteva
Joan Gispert
Lluís Godo
Franco Montagna
Carles Noguera
Title: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies
Journal: Annals of Pure and Applied Logic
Volume 160
Number 1
Pages: 53-81
Year: 2009
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This paper is a contribution to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta} -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 propose 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 (all ultraproducts of the real unit interval), the strict hyperreals (only ultraproducts giving a proper extension of the real unit interval) 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 (strict) 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.