Abstracts Vol. 9-No 1/ 1999
Next
Previous
Summaries
Special Issue: Multi-valued Logics
Guest editor: Walter Carnielli
Abstract: In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in \cite{baazcuteli} and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in (Baaz, Fermueller and Zach 1994). For the special case of the finite linear mv-algebras, the Strong Completeness Theorem was proved in (Gil, Torrens, and Verdú 1997), as a consequence of McNaughton's Theorem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems.
Keywords: Many-valued propositional logic, Gentzen system, sequent calculus, deduction theorem, completeness, cut elimination, finite algebra.
Abstract: The term fuzzy logic has two different meanings --broad and narrow. In Zadeh's opinion (Zadeh 1988), fuzzy logic (in the narrow sense) is an extension of many-valued logic but having a different agenda - as generalized modus ponens, max-min inference, linguistic quantifiers etc. The question we address in this paper is whether there is something in Zadeh's specific agenda which cannot be grasped by ``classical", ``traditional" mathematical (many-valued) logic. We show that much of fuzzy logic can be understood as classical deduction in a many-sorted many-valued Pavelka-Lukasiewicz style rational quantification logic. This means that, besides the linguistic or approximation aspects, the logical aspect (symbolic, deductive) is present too and can be made explicit.
Abstract: In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations --- classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations of the quasivarieties generated by the 12 logics involved (that prove to be varieties), we find natural equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequential axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic.
Keywords: Belnap's four-valued logic, polynomial operation, functionally complete logic, De Morgan lattice, distributive bilattice with negation, Boolean algebra, implicative De Morgan lattice, sequential consequence, equational consequence, algebraizable sequential consequence, equivalent quasivariety.
Abstract: In an unpublished paper (see Galli and Sagastume, 1996), we prove the equivalence between validity in 3L-models and algebraic validity in 3-valued Lukasiewicz algebras. R. Cignoli and M. Sagastume de Gallego present in (Cignoli and M. Sagastume de Gallego 1981) an intrinsic definition of the operators si for i=1,...,4 of a 5-valued Lukasiewicz algebra.
The aim of the present work is to study those operators in g-Kripke models context and to generalize the result obtained for 3L-models in (Galli and Sagastume, 1996) by proving that there exist g-Kripke models appropriate for 5-valued Lukasiewicz propositional calculus (see theorem 3.6). As a corollary, we find the models for 4 and 3-valued Lukasiewicz propositional calculi.
Abstract: This work presents the concepts of translation and conservative translation between logics. By using algebraic semantics we introduce several conservative translations involving the classical propositional calculus and the many-valued calculi of Post and Lukasiewicz.
Keywords: translations, conservative translation, many-valued logics, algebraic semantics, Lukasiewicz logics and algebras, Post logics and algebras.
Abstract: We characterize, for every subvariety V of the variety of all MV-algebras, the free objects in V. We use our results to compute coproducts in V and to provide simple single-axiom axiomatizations of all many-valued logics extending the Lukasiewicz one.
Abstract: In this paper we shall prove that l-rings are categorally equivalent to the MV*-algebras, a subcategory of perfect MV-algebras. We shall use this equivalence in order to characterize l-rings as quotients of certain semirings of matrices over MV*-algebras. We shall establish a relation between l-ideals in l-rings and some ideals in MV*-algebras. This allows us to study the MV*f-algebras, a subclass of the MV*-algebras corresponding to the f-rings.
Abstract: The variety O2 of double Ockham algebras consists of the algebras (A; v, ^,f,g,0,1) of type (2,2,1,1,0,0) where (A; v, ^,f,0,1) and (A; v, ^,g,0,1) are Ockham algebras. In (Sequeira 1993), M. Sequeira introduced several subvarieties of O2. In this paper we give a construction of free double Ockham algebras on a partially ordered set. We also describe free objects for the subvarieties of O2 considered in (Sequeira 1993).
Journal of Applied Non-Classical Logics
Volume 9- No 1/ 1999
Next
Previous
Summaries