Hoops and Fuzzy Logic

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Francesc Esteva
Lluís Godo
Petr Hájek
Franco Montagna
Title: Hoops and Fuzzy Logic
Journal: Journal of Logic and Computation
Volume 13
Number 4
Pages: 531-555
Year: 2003


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 falsegood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom 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 (\forall 3)} in these predicate logics is studied. Finally, computational complexity issues of the propositional logics are also addressed.