The logic of distributive bilattices

Bou, Félix y Rivieccio, Umberto . (2011) The logic of distributive bilattices. Logic Journal of the IGPL, 19 (1), 2011, p. 183-216.

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Título The logic of distributive bilattices
Autor(es) Bou, Félix
Rivieccio, Umberto
Materia(s) Filosofía
Abstract Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú (1995, Algebraizable Gentzen Systems and the Deduction of Theorem for Gentzen Systems) (even if the logic itself is not algebraizable). We also prove some purely algebraic results concerning bilattices, for instance that the variety of (unbounded) distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron (1996, Math. Struct. Comput. Sci., 6, 287–299) stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices (of which distributive bilattices are a proper subclass).
Palabras clave Bilattice
many-valued logic
abstract algebraic logic
reduced model
non-protoalgebraic logic
algebraizable Gentzen system.
Editor(es) Oxford University Press
Fecha 2011
Formato application/pdf
Identificador bibliuned:81-Urivieccio-0002
http://e-spacio.uned.es/fez/view/bibliuned:81-Urivieccio-0002
DOI - identifier https://doi.org/10.1093/jigpal/jzq041
ISSN - identifier 1367-0751 - eISSN 1368-9894
Nombre de la revista Logic Journal of the IGPL
Número de Volumen 19
Número de Issue 1
Página inicial 183
Página final 216
Publicado en la Revista Logic Journal of the IGPL, 19 (1), 2011, p. 183-216.
Idioma eng
Versión de la publicación acceptedVersion
Tipo de recurso Article
Derechos de acceso y licencia http://creativecommons.org/licenses/by-nc-nd/4.0
info:eu-repo/semantics/openAccess
Tipo de acceso Acceso abierto
Notas adicionales The registered version of this article, first published in Logic Journal of the IGPL, is available online at the publisher's website: Oxford University Press, https://doi.org/10.1093/jigpal/jzq041
Notas adicionales La versión registrada de este artículo, publicado por primera vez en Logic Journal of the IGPL, está disponible en línea en el sitio web del editor: Oxford University Press, https://doi.org/10.1093/jigpal/jzq041

 
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