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The logic of distributive bilattices

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dc.contributor.authorBou, Félix
dc.contributor.authorRivieccio, Umberto
dc.date.accessioned2024-05-21T12:53:21Z
dc.date.available2024-05-21T12:53:21Z
dc.date.issued2011
dc.description.abstractBilattices, 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).en
dc.description.versionversión final
dc.identifier.doihttps://doi.org/10.1093/jigpal/jzq041
dc.identifier.issn1367-0751 - eISSN 1368-9894
dc.identifier.urihttps://hdl.handle.net/20.500.14468/19441
dc.journal.issue1
dc.journal.titleLogic Journal of the IGPL
dc.journal.volume19
dc.language.isoen
dc.publisherOxford University Press
dc.relation.centerFacultad de Filosofía
dc.relation.departmentLógica, Historia y Filosofía de la Ciencia
dc.rightsinfo:eu-repo/semantics/openAccess
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
dc.subject.keywordsBilattice
dc.subject.keywordsmany-valued logic
dc.subject.keywordsabstract algebraic logic
dc.subject.keywordsreduced model
dc.subject.keywordsnon-protoalgebraic logic
dc.subject.keywordsalgebraizable Gentzen system.
dc.titleThe logic of distributive bilatticeses
dc.typejournal articleen
dc.typeartículoes
dspace.entity.typePublication
relation.isAuthorOfPublication78477d31-191f-4cbb-b9ff-32b8ec63d72b
relation.isAuthorOfPublication.latestForDiscovery78477d31-191f-4cbb-b9ff-32b8ec63d72b
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