Free three-valued Closure Lukasiewicz Algebras

In this paper, the structure of finitely generated free objects in the variety of three-valued closure Lukasiewicz algebras is determined. We describe their indecomposable factors and we give their cardinality.Fil: Abad, Manuel. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Díaz Varela, José Patricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Rueda, Laura Alicia. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Suardíaz, Ana María. Universidad Nacional del Sur. Departamento de Matemática; Argentin

Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work

On the relatively Pseudo-complement operation in finite RHAs

Refined hedge algebras were introduced and  investigated by Ho Nam in [6--9]. It is known [9] that every refined hedge algebra (RHA, for short) with a chair of the primary generators is a distributive lattice. In this paper we restrict our consideration to finite version of RHAs (see [7, 9]). It is shown that every finite RHA is a Heyting (pseudo-Boolean) algebra. Furthermore, some computing results for the relatively pseudo-complement operation in  these algebras will be exhibited

Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work

Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

On the notion of negation in certain non-classical propositional logics

The purpose of this study is to investigate some aspects of how negation functions in certain non-classical propositional logics. These include the intuitionistic system developed by Heyting, the minimal calculus proposed by Johansson, and various intermediate logics between the minimal and the classical systems. Part I contains the new results which can be grouped into two classes: extension-criteria results and infinite chain results. In the first group criteria are given for answering the question: when do formulae added to the axioms of the minimal calculus as extra axioms extend the minimal calculus to various known intermediate logics? One of the results in this group (THEOREM 1 in Chapter II, Section 1) is a generalization of a result of Jankov. In the second group certain intermediate logics are defined which form infinite chains between well-known logical systems. One of the results here (THEOREM 1 in Chapter II, Section 2) is a generalization of a result of McKay. In Part II the new results are discussed from the viewpoint of negation. It is rather difficult, however, to draw definite conclusions which are acceptable to all. For these depend on, and are closely bound up with, certain basic philosophical presuppositions which are neither provable, nor disprovable in a strict sense. Taking an essentially classical position, it is argued that the logics appearing in the defined infinite chains are such that they diverge only in the vicinity of negation, and the notions of negation in them are simply ordered in a sense which is specified during the discussion. In Appendix I a number of conjectures are formulated in connection with the new results.<p