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OUP > Journals > Computing/Engineer. & Mathematics/Stats. > Journal of Logic and Computation

Journal of Logic and Computation

Volume 13, Issue 4, June 2003: pp. 532-555

Original Article
Hoops and Fuzzy Logic

Francesc Esteva1, Lluís Godo1, Petr Hájek2 and Franco Montagna3

1Institut d' Investigació en Intelligència Artificial - CSIC, Campus Univ. Autònoma de Barcelona s/n, 08193 Bellaterra, Spain. E-mail: esteva@iiia.csic.es, godo@iiia.csic.es
2Institute of Computer Science, Academy of Sciences, 182 07 Prague, Czech Republic. E-mail: hajek@cs.cas.cz
3Dipartamento de Matematica, Università degli Studi di Siena, Via del Capitano 15, 53100 Siena, Italy. E-mail: montagna@unisi.it
Received 21 January 2002

In this paper we investigate the falsehood-free fragments of main residuated fuzzy logics related to continuous t-norms (Hájek's Basic fuzzy logic BL and some well-known axiomatic extensions), and we relate them to the varieties of 0-free subreducts of the corresponding algebras. These turn out to be classes of algebraic structures known as hoops. We provide axiomatizations of all these fragments and we call them hoop logics; we prove they are strongly complete with respect to their corresponding classes of hoops, and that each fuzzy logic is a conservative extension of the corresponding hoop logic. Analogously, we also study the falsehood-free fragment of a weaker logic than BL, called MTL, which is the logic of left-continuous t-norms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom ([forall]3) in these predicate logics is studied. Finally, computational complexity issues of the propositional logics are also addressed.

Mathematical fuzzy logics, BL-algebras, falsehood-free fragments, hoops, conservativeness.

In this paper we investigate the falsehood-free fragments of main residuated fuzzy logics related to continuous t-norms (Hájek's Basic fuzzy logic BL and some well-known axiomatic extensions), and we relate them to the varieties of 0-free subreducts of the corresponding algebras. These turn out to be classes of algebraic structures known as hoops. We provide axiomatizations of all these fragments and we call them hoop logics; we prove they are strongly complete with respect to their corresponding classes of hoops, and that each fuzzy logic is a conservative extension of the corresponding hoop logic. Analogously, we also study the falsehood-free fragment of a weaker logic than BL, called MTL, which is the logic of left-continuous t-norms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom ([forall]3) in these predicate logics is studied. Finally, computational complexity issues of the propositional logics are also addressed.

Keywords: Mathematical fuzzy logics, BL-algebras, falsehood-free fragments, hoops, conservativeness.

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