Fuzzy Sets and Systems 158 (2007) 2575 – 2590 www.elsevier.com/locate/fss On three implication-less fragments of t-norm based fuzzy logics Romà Adillon a , Àngel García-Cerdaña b, c , Ventura Verdú d , ∗ a Departament de Matemàtica Econòmica, Financera i Actuarial, Universitat de Barcelona, Spain b Institut d’Investigació en Intel.ligència Artificial, CSIC, Bellaterra, Spain c Departament de Filosofia, Universitat Autònoma de Barcelona, Spain d Departament de Probabilitat, Lògica i Estadística,Universitat de Barcelona, Spain Received 2 November 2005; received in revised form 6 March 2007; accepted 6 March 2007 Available online 19 March 2007 Abstract This paper concerns the study of the fragments without implication of the logic of residuated lattices (or monoidal logic, or intuitionistic logic without contraction). We obtain that these fragments are exactly the same fragments as those found in classical logic. As a corollary of this result we obtain that the implication-less fragments of every t-norm based fuzzy logic are exactly the same fragments as those found in classical logic. © 2007 Elsevier B.V. All rights reserved. Keywords: Substructural logics; t-Norm based logics; Residuated lattices; Semilatticed monoids; Gentzen systems 1. Introduction Esteva and Godo defined [12] the logic MTL (monoidal t-norm based logic), a generalization of BL, the basic fuzzy Logic defined by Hájek [18] which is the logic of continuous t-norms and their residua [9]. It was proved in [20] that MTL is the logic of left-continuous t-norms and their residua, that is, the logic of the class of the (commutative integral bounded) residuated lattices defined in the real unit interval [0, 1]. The word ‘monoidal’ in the name of this logic comes from the fact that MTL can be seen as an axiomatic extension of the so-called monoidal logic considered by Höhle in [19], where a completeness theorem with respect to the class of all residuated lattices is proved. Monoidal logic is definitionally equivalent to other systems considered in the literature: H BCK [28] (corresponding to what Ono now calls FL ew -logic), IPC ∗ \c [1] (intuitionistic logic without contraction), etc. Beyond the connection of monoidal logic with residuated lattices by means of a completeness theorem, we can also say that monoidal logic is the logic of residuated lattices in a stronger sense: it was proved in [1] that this logic is algebraizable in the sense of [5] and the variety of residuated lattices is its equivalent algebraic semantics. In [16] (see also [17]) it was shown that MTL can be obtained as the extension of the monoidal logic with the prelinearity axiom ( → ) ∨ ( → ). Thus, the study of the logic of residuated lattices is important in the context of the studies of the t-norm based fuzzy logics [18] because ∗ Corresponding author. Facultat de Matematiques, University of Barcelona, Gran Via 585 08007 Barcelona, Spain. Tel.: +34934021658; fax: +34 934021601. E-mail addresses: adillon@ub.edu (R. Adillon), angel@iiia.csic.es, Angel.Garcia.Cerdana@uab.es (À. García-Cerdaña), v.verdu@ub.edu (V. Verdú). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.03.008