TitleSobre lógicas fuzzy basadas en T–normas y los resultados de Monteiro sobre álgebras de Heyting simétricas
Publication TypeConference Paper
Year of Publication2007
AuthorsEsteva F, Godo L
Conference NameActas del IX Congreso Dr. António A. R. Monteiro
Conference LocationBahía Blanca (Argentina)
Date Published30/05/2007

In the setting of logical systems associated to residuated lattices, Ono defined in [20] (see also [7]) the weak contractive systems corresponding to pseudo-complemented residuated lattices. Intuitionistic logic (associated to Heyting algebras) and the infinitely- valued Gödel logic (associated to linear Heyting algebras) are remarkable examples of weak contractive systems. In these systems, the definable negation $\neg\varphi$ as $\varphi \to 0$ is not involutive (it is in fact Gödel negation over linearly ordered algebras). In this setting, it makes sense to study the issue of adding an involutive negation to weak contractive systems. This is in fact what Monteiro did in his excellent monograph [18] for the case of Heyting algebras, the resulting structures being called Symmetric Heyting algebras, and paying special attention to the linear case. In this short paper, after a general presentation of the t-norm based (fuzzy) logics as pre-linear extensions of Höhle’s Monoidal logic [15] or Ono’s FLew, we will study similar expansions for any (prelinear) weak contractive fuzzy logic, following the line of the paper [9] where the authors studied the expansions obtained by adding an involutive negation to SBL, Gödel and Product logics. This line has been further investigated in [4] and [11]. We show that some of Monteiro’s results are also valid in the general setting of prelinear pseudocomplemented residuated lattices and lead to new axiomatic presentations.