OUP > Journals > Computing/Engineer. & Mathematics/Stats. > Journal of Logic and Computation
Journal of Logic and Computation
Volume 13, Issue 4, June 2003: pp. 532555
Original Article Hoops and Fuzzy Logic
Francesc Esteva^{1}, Lluís Godo^{1}, Petr Hájek^{2} and Franco Montagna^{3}
^{1}Institut d' Investigació en Intelligència Artificial  CSIC, Campus Univ. Autònoma de Barcelona s/n, 08193 Bellaterra, Spain. Email: esteva@iiia.csic.es, godo@iiia.csic.es
^{2}Institute of Computer Science, Academy of Sciences, 182 07 Prague, Czech Republic. Email: hajek@cs.cas.cz
^{3}Dipartamento de Matematica, Università degli Studi di Siena, Via del Capitano 15, 53100 Siena, Italy. Email: montagna@unisi.it
Received 21 January 2002
In this paper we investigate the falsehoodfree fragments of main residuated fuzzy logics related to continuous tnorms (Hájek's Basic fuzzy logic BL and some wellknown axiomatic extensions), and we relate them to the varieties of 0free 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 falsehoodfree fragment of a weaker logic than BL, called MTL, which is the logic of leftcontinuous tnorms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehoodfree 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, BLalgebras, falsehoodfree fragments, hoops, conservativeness.
In this paper we investigate the falsehoodfree fragments of main residuated fuzzy logics related to continuous tnorms (Hájek's Basic fuzzy logic BL and some wellknown axiomatic extensions), and we relate them to the varieties of 0free 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 falsehoodfree fragment of a weaker logic than BL, called MTL, which is the logic of leftcontinuous tnorms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehoodfree 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, BLalgebras, falsehoodfree fragments, hoops, conservativeness.
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