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Journal of Logic and Computation Advance Access published online on May 9, 2008
Journal of Logic and Computation, doi:10.1093/logcom/exn015
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

Logical Weak Completions of Paraconsistent Logics

Mauricio Osorio Galindo

Universidad de las Américas – Puebla E-mail: osoriomauri{at}gmail.com

José R. Arrazola Ramírez and José Luis Carballido

Benemérita Universidad Autónoma de Puebla, Mathematics Department E-mail: arrazola{at}fcfm.buap.mx, carballido{at}fcfm.buap.mx

Received 25 March 2008.


  Abstract

Let P be an arbitrary theory and let X be any given logic. Let M be a set of atoms. We say that M is a X-stable model of P if M is a classical model of P and P{cup}¬Formula proves in logic X all atoms in M, this is denoted by P{cup}¬Formula {Vdash}xM. We prove that being an X-stable model is an invariant property for disjunctive programmes under a large class of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that replaces ¬a with ¬{square}a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning are equivalent for disjunctive programmes. In addition, we show that such semantics can be characterized by a fixed-point operator in terms of classical logic. We also present a simple translation of a disjunctive programme D into a normal programme N, such that the PStable model semantics of N corresponds to the stable semantics of D over the common language. We present the formal proof of this statement.

Keywords: Paraconsistent logics; multivalued logics; non-monotonic semantics; stable semantics


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