Fidel Semantics for Propositional and First-Order Version of the Logic of CG’3
DOI:
https://doi.org/10.12775/LLP.2022.019Keywords
paraconsistent logics, first-order logics, Fidel semanticsAbstract
Paraconsistent extensions of 3-valued Gödel logic are studied as tools for knowledge representation and nonmonotonic reasoning. Particularly, Osorio and his collaborators showed that some of these logics can be used to express interesting nonmonotonic semantics. CG’3 is one of these 3-valued logics. In this paper, we introduce Fidel semantics for a certain calculus of CG’3 by means of Fidel structures, named CG’3-structures. These structures are constructed from enriched Boolean algebras with a special family of sets. Moreover, we also show that the most basic CG’3-structures coincide with da Costa–Alves’ bi-valuation semantics; this connection is displayed through a Representation Theorem for CG’3-structures. By contrast, we show that for other paraconsistent logics that allow us to present semantics through Fidel structures, this connection is not held. Finally, Fidel semantics for the first-order version of the logic of CG’3 are presented by means of adapting algebraic tools.
References
Burris, S., and H. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78, Springer, 1981.
Béziau, J-Y., “The paraconsistent logic Z: A possible solution to Jaśkowski’s problem”, Logic and Logical Philosophy, 15, 2 (2006): 99–111. DOI: http://dx.doi.org/10.12775/LLP.2006.006
Borja-Macías, V., and M. Pérez-Gaspar, “Kripke-type semantics for CG’3”, Electronic Notes in Theoretical Computer Science, 328 (2016): 17–29.
Carnielli, W., and M. Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, vol. 40, Logic, Epistemology, and the Unity of Science, Basel, Switzerland: Springer International Publishing, 2016. DOI: http://dx.doi.org/10.1007/978-3-319-33205-5
Carnielli, W. A., M. E. Coniglio, R. Podiacki and T. Rodrigues, “On the way to a wider model theory: Completeness theorems for first-order logics of formal inconsistency”, The Review of Symbolic Logic, 7, 3 (2014): 548–578. DOI: http://dx.doi.org/10.1017/S1755020314000148
Ciuciura, J., “Paraconsistency and Sette’s calculus P1”, Logic and Logical
Philosophy, 24 (2015): 265–273. DOI: http://dx.doi.org/10.12775/LLP.2015.003
Ciuciura, J., “Sette’s calculus P1 and some hierarchies of paraconsistent systems”, Journal of Logic and Computation, 30, 5 (2020): 1109–1124.
Ciuciura, J., “On the system CB1 and a lattice of the paraconsistent calculi”, Logic and Logical Philosophy, 29 (2020): 223–237. DOI: http://dx.doi.org/10.12775/LLP.2019.035
Coniglio, M., Figallo-Orellano, A., Hernández-Tello, A. and M. Pérez-Gaspar, “CG’3 as the logic of modal 3-valued Heyting algebras”, Journal of Applied Logic, 9, 1 (2022): 175–197.
da Costa, N., “On the theory of inconsistent formal systems”, Notre Dame Journal of Formal Logic, 15, 4 (1974): 497–510. DOI: http://dx.doi.org/10.1305/ndjfl/1093891487
da Costa, N., and E. Alves, “A semantical analysis of the calculi Cn”, Notre Dame Journal of Formal Logic, 18 (1977): 621–30.
D’Ottaviano, I., “The model extension theorems for J3-theories”, pages 157–173 in Methods in Mathematical Logic, Proceedings of the 6th Latin American Symposium on Mathematical Logic, 1130, Lecture Notes in Mathematics, Springer, Berlin, 1985.
D’Ottaviano, I., “Definability and quantifier elimination for J3-theories”, Studia Logica, 46 (1987): 37–54.
Fidel, M. “The decidability of the calculi Cn”, Reports on Mathematical Logic, 8 (1977): 31–40.
Figallo-Orellano, A., and J. Slagter, “An algebraic study of the first order version of some implicational fragments of the three-valued Łukasiewicz logic”, Computación y Sistemas (2022).
Figallo-Orellano, A., and J. Slagter, “Monteiro’s algebraic notion of maximal consistent theory for Tarskian logics”, Fuzzy Sets and Systems (2022). DOI: http://dx.doi.org/10.1016/j.fss.2022.04.007
Figallo-Orellano, A., M. Pérez-Gaspar J. M. Ramírez-Contreras, “Paraconsistent and paracomplete logics based on k-cyclic modal pseudocomplemented De Morgan algebras”, Studia Logica (2022).
Fitting, M., Intuitionistic Logic, Model Theory and Forcing, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam–London, 1969.
Loparić, A., “A semantical study of some propositional calculi”, The Journal of Non-Classical Logic, 3 (1986): 73–95.
Mendelson, E., Introduction to Mathematical Logic, CRC Press, 2009.
Monteiro, A., “Sur les algèbres de Heyting simetriques”, Portugaliae Math., 39, 1–4 (1980): 1–237.
Martinez, G., “Elimination of quantifiers on Łukasiewicz logics”, Zeitschr. f. math. Logik und Grundlagen d. Math., 35 (1989): 15–21.
Martinez, G., “A model theory based on Łukasiewicz logics” (in Spanish), Master Thesis, Universidad Nacional del Sur, 1989.
Odintsov, S., “Algebraic semantics for paraconsistent Nelson’s logic”, Journal of Logic and Computation, 13, 4 (2003).
Osorio, M., and J. Carballido, “Brief study of G’3 logic”, Journal of Applied Non-Classical Logics, 18, 4 (2008): 475–499.
Osorio, M., J. Carballido and C. Zepeda, “Revisiting Z”, Notre Dame Journal of Formal Logic, 55, 1 (2014): 129–155. DOI: http://dx.doi.org/10.1215/00294527-2377905
Osorio, M., A. Figallo-Orellano and M. Pérez-Gaspar, “A family of genuine and non-algebraizable C-systems”, Journal of Applied Non-Classical Logics, 31, 1 (2021): 56—84.
Osorio, M., J. Navarro, J. Arrazola and V. Borja, “Logics with common weak completions”, Journal of Logic and Computation, 16, 6 (2006): 867–890.
Pearce, D., “Stable inference as intuitionistic validity”, Logic Programming, 38 (1999): 79–91.
Pérez-Gaspar, M., A. Hernández-Tello, J. Arrazola-Ramírez and M. Osorio, “An axiomatic approach to CG’3 logic”, Logic Journal of the IGPL, 28, 6 (2020): 1218–1232.
Wójcicki, R., Lectures on Propositional Calculi, Warsaw, 1984.
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