Functional Definability Issues in Logics Based on Triangular Norms

Enrico Marchioni Affiliation: Universitat Oberta de Catalunya. Departamento de Ciencias de la Información y la Comunicación (Barcelona, España) Biography: Not available Close window

Enrico Marchioni About the authors  Publication year: 2007 Language: English Subjects: Science and Technology Collection: Monografies de l'Institut d'Investigació en Intel-ligencia Artificial Free eBook

Abstract:

The last decade has seen a tremendous development in infinitely valued logics which take the real unit interval as the basic set of truth values. This set is usually endowed with an algebraic structure of residuated lattice defined by a commutative semigroup operation –particularly a (left) continuous t-norm– together with its residuum, which are interpreted as a non-idempotent conjunction and implication operations respectively. These logics, commonly referred to as t-norm based fuzzy logics, are at the core of a new emerging discipline which is named mathematical fuzzy logic, after Petr Hájek. Since continuous t-norms are ordinal sums of isomorphic copies of the Lukasiewicz t-norm, the Product t-norm and the minimum operation, the traditional infinitely valued Lukasiewicz and Gödel logics, together with the Product logic, form the core examples for such mathematical fuzzy logics. One stream of the development in mathematical fuzzy logic is towards enhancing the expressive power of these logics by adding new connectives. Of particular interest are the systems obtained from the addition of an independent involutive negation or by considering a combination of the Lukasiewicz and Product systems, which results in a conservative extension of both systems, and that, additionally, has the Gödel logic as a subsystem. Both topics (among others) are addressed by the author of this monograph, which is based on his Ph.D. dissertation.

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