Quasi-N4-lattices and their logic

• Clodomir Silva Lima Neto UFRN
• Thiago Nascimento da Silva Cebraspe
• Umberto Rivieccio UFRN

DOI: https://doi.org/10.5753/wbl.2022.222852

Resumo

The variety of quasi-N4-lattices (QN4) was recently introduced as a non-involutive generalization of N4-lattices (algebraic models of Nelson's paraconsistent logic). While research on these algebras is still at a preliminary stage, we know that QN4 is an arithmetical variety which possesses a ternary as well as a quaternary deductive term, enjoys equationally definable principal congruences and the strong congruence extension property. We furthermore have recently introduced an algebraizable logic having QN4 as its equivalent semantics. In this contribution we report on the results obtained so far on this class of algebras and on its logical counterpart.

Palavras-chave: Non-involutive, Paraconsistent Nelson, Algebraizable logic

Referências

Almukdad, A. and Nelson, D. (1984). Constructible falsity and inexact predicates. JSL, 49(1):231–233.

Blok, W. J. and Pigozzi, D. (1994). On the structure of varieties with equationally definable principal congruences III. Algebra Universalis, 32:545–608.

Burris, S. and Sankappanavar, H. P. (2012). A Course in Universal Algebra. Springer, The Millenium edition.

Font, J. M. (2016). Abstract Algebraic Logic: An Introductory Textbook. Imperial College Press.

Galatos, N. and Raftery, J. G. (2015). Idempotent Residuated Structures: Some Category Equivalences and their Applications. Transactions of the American Mathematical Society, 367(5):3189–3223.

Liang, F. and Nascimento, T. (2019). Algebraic Semantics for Quasi-Nelson Logic. In Iemhoff, R., Moortgat, M., and de Queiroz, R., editors, Logic, Language, Information, and Computation, volume 11541, pages 450–466, Berlin, Heidelberg. Springer Berlin Heidelberg.

Lima Neto, C. S., Nascimento, T., and Rivieccio, U. (2022). Algebraizability of the Logic of Quasi-N4-Lattices. In Indrzejczak, A. and Zawidzki, M., editors, 10th International Conference on Non-Classical Logics. Theory and Applications (NCL 2022), pages 240–253, University of ódz, Poland.

Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14(1):16–26.

Odintsov, S. P. (2003). Algebraic Semantics for Paraconsistent Nelson’s Logic. Journal of Logic and Computation, 13(4):453–468.

Odintsov, S. P. (2004). On the Representation of N4-Lattices. Studia Logica, 76(3):385–405.

Rivieccio, U. (2020). Fragments of Quasi-Nelson: two negations. Journal of Applied Logic, 7(4):499–559.

Rivieccio, U. (2021). Fragments of Quasi-Nelson: The Algebraizable Core. Logic Journal of the IGPL.

Rivieccio, U. (2022). Quasi-N4-lattices. Soft Computing, 26:2671–2688.

Rivieccio, U. and Jansana, R. (2021). Quasi-Nelson algebras and fragments. Mathematical Structures in Computer Science, 31(3):257–285.

Rivieccio, U. and Spinks, M. (2021). Quasi-Nelson; Or, Non-involutive Nelson Algebras. In Fazio, D., Ledda, A., and Paoli, F., editors, Algebraic Perspectives on Substructural Logics, volume 55 of Trends in Logic, pages 133–168. Springer International Publishing.

Sendlewski, A. (1990). Nelson Algebras through Heyting ones: I. Studia Logica: An International Journal for Symbolic Logic, 49(1):105–126.

Spinks, M. and Veroff, R. (2018). Paraconsistent constructive logic with strong negation as a contraction-free relevant logic. In Czelakowski, J., editor, Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science, volume 16 of Outstanding Contributions to Logic, pages 323–379. Springer International Publishing.