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2022-10-01
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info:eu-repo/semantics/openAccess
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Oxford University Press
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Resumen
This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic
(full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a.
-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.
Descripción
The registered version of this article, first published in “Logic Journal of the IGPL, Volume 30, 2022", is available online at the publisher's website: Oxford University Press, https://doi.org/10.1093/jigpal/jzab023
La versión registrada de este artículo, publicado por primera vez en “Logic Journal of the IGPL, Volume 30, 2022", está disponible en línea en el sitio web del editor: Oxford University Press, https://doi.org/10.1093/jigpal/jzab023
Financiado por: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil), under the grant 313643/2017-2
La versión registrada de este artículo, publicado por primera vez en “Logic Journal of the IGPL, Volume 30, 2022", está disponible en línea en el sitio web del editor: Oxford University Press, https://doi.org/10.1093/jigpal/jzab023
Financiado por: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil), under the grant 313643/2017-2
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Umberto Rivieccio, Fragments of Quasi-Nelson: The Algebraizable Core, Logic Journal of the IGPL, Volume 30, Issue 5, October 2022, Pages 807–839, https://doi.org/10.1093/jigpal/jzab023
Centro
Facultad de Filosofía
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Lógica, Historia y Filosofía de la Ciencia