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A. Rose, J. Rosser (1958)
Fragments of many-valued statement calculiTransactions of the American Mathematical Society, 87
F. Montagna (2000)
Review: Petr Hájek, Lluis Godo, Francesc Esteva, A Complete Many-Valued Logic with Product-ConjunctionThe Bulletin of Symbolic Logic, 6
(1996)
Innnite-valued GG odel logics with 0-1 projections and relativizations . In G ODEL'96 | Logical foundations of mathematics, computer science and physics
(1997)
Svejda : A strong completeness theorem for nitelyaxiomatized fuzzy theories
P. Hájek (1998)
Metamathematics of Fuzzy Logic, 4
0 is a cluster point of E( * )), or 2. there is an interval I 0 = [0, a] ∈ I(E( * )) and * I 0 is isomorphic to the product t-norm
Proposition 1 A continuous t-norm has non-trivial zero divisors ii it is an ordinal sum such that there exists I 0 = 0; a] 2 I(E()) and I0 is isomorphic to Lukasiewicz t-norm
G. Takeuti, Satoko Titani (1992)
Fuzzy logic and fuzzy set theoryArchive for Mathematical Logic, 32
Theorem 16 If I denotes the restriction of a continuous t-norm to I I, I 2 I(E()), then: 1. For each I 2 I(E()), I is isomorphic either to the product t-norm (on 0; 1]) or to Lukasiewicz t-norm
A. Turquette (1959)
Review: Alan Rose, J. Barkley Rosser, Fragments of Many-valued Statement CalculiJournal of Symbolic Logic, 24
If x; y 2 0; 1] are such that there is no I 2 I(E()) with x; y 2 I, then x y = min(x; y)
M. Baaz (1996)
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M. Dummett (1959)
A propositional calculus with denumerable matrixJournal of Symbolic Logic, 24
P. Hájek, L. Godo, F. Esteva (1996)
A complete many-valued logic with product-conjunctionArchive for Mathematical Logic, 35
P. Hájek (1995)
Fuzzy logic and arithmetical hierarchyFuzzy Sets and Systems, 73
Then the following representation theorem, due to Ling, for continuous t-norms holds (see 4] for a proof
John Bacon (1968)
Review: Michael Dummett, A Propositional Calculus with Denumerable MatrixJournal of Symbolic Logic, 33
Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation.
Archive for Mathematical Logic – Springer Journals
Published: Feb 1, 2000
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