Algebraic study of axiomatic extensions of triangular norm based fuzzy logics

According to the Zadeh’s famous distinction, Fuzzy Logic in narrow sense, as opposed to Fuzzy Logic in broad sense, is the study of logical systems aiming at a formalization of approximate reasoning. In the systems commonly used the strong conjunction connective is interpreted by a triangular norm (t-norm, for short) while the implication connective is interpreted by its residuum. Therefore, the usual logical systems for Fuzzy Logic are based on t-norms with a residuum. The necessary and sufficient condition for a t-norm to have a residuum is the left-continuity. In order to define the based t-norm based fuzzy logic, Esteva and Godo introduced the system MTL, which was indeed proved to be complete with respect to the semantics given by all left-continuous t-norms and their residua. In the first part of this dissertation we have carried out an attempt to describe the axiomatic extensions of MTL, paying special attention to those which are also t-norm based. We have done it from an algebraic point of view, by exploiting the fact that these logics are algebraizable by varieties of MTL-algebras. Therefore, our study has resulted in an algebraic study of such varieties, where the final aim would be to obtain a description of the structure of their lattice and their relevant properties. Although this description has not been achieved yet, we have done several significant advances in this direction that can be classified in two groups: (a) those that spread some light over the amazing complexity of the lattice, and (b) those that describe some well-behaved parts of the lattice. More precisely: • By considering the connected rotation-annihilation method proposed to build involutive left-continuous continuous t-norm, we have proposed a possible way to decompose MTL-chains and we have studied some particular cases of this decomposition. This has resulted in an extension of the theory of perfect, local and bipartite algebras formerly used in varieties of MV and BL-algebras, to the variety of all MTL-algebras. • Perfect IMTL-algebras have been proved to be exactly (module isomorphism) the disconnected rotations of prelinear semihoops (a particular case of the decomposition as connected rotation-annihilation). • The lattice of varieties generated by perfect IMTL-algebras has been proved to be isomorphic to the lattice of varieties of prelinear semihoops. xiii xiv • A decomposition theorem of every MTL-chain as an ordinal sum of indecomposable prelinear semihoops has been obtained. Since all IMTL-chains are indecomposable and, as the previous item states, we have the complexity of all the lattice of varieties inside the involutive part, the description of all indecomposable prelinear semihoops seems to be a hopeless task. • A particular class of indecomposable MTL-chains has been studied, namely weakly cancellative chains. We have studied the logics associated to these chains. • We have studied the varieties of MTL-chains where a weak form of contraction, the so-called n-contraction law, holds. This condition yields a global form of Deduction Detachment Theorem and allows to prove several properties of their related logics. • We have focused on a particular subvariety of 3-contractive MTL-algebras, namely Weak Nilpotent Minimum algebras, obtaining a number of results on axiomatization of their subvarieties, local finiteness, generic chains and standard completeness. In the second part of the dissertation we consider another significant question of Fuzzy Logic: which should be the use of the intermediate truth-values? In MTL and its extensions, these truth-values for partial truth do not seem to be used in a deep way, since in the algebraization of the logics the only distinguished value is the top element. Following Pavelka’s idea, we consider expansions of tnorm based logics with constants for the intermediate truth-values, allowing them to play an explicit role in the language. The originality of our proposal lies in carrying out an algebraic approach to these expansions and studying their standard completeness properties