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SOME MODEL-THEORETIC RESULTS ON THE 3-VALUED PARACONSISTENT FIRST-ORDER LOGIC QCIORE

Part of: Model theory General logic

Published online by Cambridge University Press:  09 December 2019

MARCELO E. CONIGLIO ,
G.T. GOMEZ-PEREIRA  and
MARTÍN FIGALLO
MARCELO E. CONIGLIO
Affiliation:
INSTITUTE OF PHILOSOPHY AND THE HUMANITIES (IFCH), AND CENTRE FOR LOGIC, EPISTEMOLOGY, AND THE HISTORY OF SCIENCE (CLE) UNIVERSITY OF CAMPINAS (UNICAMP) R. CORA CORALINA, 100 - CAMPINAS - SP, 13083-896 - BRAZILE-mail: [email protected]
G.T. GOMEZ-PEREIRA
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDAD NACIONAL DEL SUR (UNS)BAHÍA BLANCA, ARGENTINAE-mail: [email protected]
MARTÍN FIGALLO
Affiliation:
INSTITUTO DE MATEMÁTICA (INMABB) DEPARTAMENTO DE MATEMÁTICA UNIVERSIDAD NACIONAL DEL SUR (UNS) - CONICETBAHÍA BLANCA, ARGENTINAE-mail: [email protected]
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Abstract

The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

Keywords

paraconsistent logicsfirst-order logicsthree-valued logicslogics of formal inconsistencymodel theory

MSC classification

Primary: 03B53: Paraconsistent logics
Secondary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 1 , March 2021 , pp. 187 - 224
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Copyright
© Association for Symbolic Logic, 2019

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References

BIBLIOGRAPHY

Béziau, J.-Y. (1990). Logiques construites suivant les méthodes de da Costa. I. Logiques paraconsistantes, paracompletes, non-alèthiques construites suivant la première méthode de da Costa (in French). Logique et Analyse (N.S.), 131/132, 259–272.Google Scholar
Carnielli, W. A. & Coniglio, M. E. (2016). Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science, Vol. 40. Cham: Springer.Google Scholar
Carnielli, W. A., Coniglio, M. E., & Marcos, J. (2007). Logics of formal inconsistency. In Gabbay, D. M. and Guenthner, F., editors. Handbook of Philosophical Logic, Second Edition, Vol. 14. Dordrecht: Springer, pp. 1–93. Doi: 10.1007/978-1-4020-6324-4_1.Google Scholar
Carnielli, W. A., Coniglio, M. E., Podiacki, R., & Rodrigues, T. (2014). On the way to a wider model theory: Completeness theorems for first-order logics of formal inconsistency. The Review of Symbolic Logic, 7(3), 548–578. Doi: 10.1017/S1755020314000148.CrossRefGoogle Scholar
Carnielli, W. A. & Marcos, J. (2002). A taxonomy of C-systems. In Carnielli, W. A., Coniglio, M. E., and D’Ottaviano, I. M. L., editors. Paraconsistency: The Logical Way to the Inconsistent. Proceedings of the 2nd World Congress on Paraconsistency (WCP 2000). Lecture Notes in Pure and Applied Mathematics, Vol. 228. New York: Marcel Dekker, pp. 1–94.CrossRefGoogle Scholar
Carnielli, W. A., Marcos, J., & de Amo, S. (2000). Formal inconsistency and evolutionary databases. Logic and Logical Philosophy, 8, 115–152.CrossRefGoogle Scholar
Cignoli, R. (1986). The class of Kleene algebras satisfying an interpolation property and Nelson algebras. Algebra Universalis, 23, 262–292.CrossRefGoogle Scholar
Coniglio, M. E., Figallo-Orellano, A., & Golzio, A. C. (2018). Non-deterministic algebraization of logics by swap structures. Logic Journal of the IGPL, to appear. First published online: November 29, 2018. Preprint available at arXiv:1708.08499 [math.LO], 2017. Doi: 10.1093/jigpal/jzy072.Google Scholar
Coniglio, M. E. & Silvestrini, L. H. (2014). An alternative approach for quasi-truth. Logic Journal of the IGPL, 22(2), 387–410. Doi: 10.1093/ljigpal/jzt026.CrossRefGoogle Scholar
da Costa, N. C. A. (1963). Sistemas formais inconsistentes (Inconsistent formal systems, in Portuguese). Habilitation Thesis, Universidade Federal do Paraná, Curitiba, Brazil. Republished by Editora UFPR, Curitiba, Brazil,1993.Google Scholar
da Costa, N. C. A. (1974). On the theory of inconsistent formal systems (Lecture delivered at the First Latin-American Colloquium on Mathematical Logic, held at Santiago, Chile, July 1970). Notre Dame Journal of Formal Logic, 15(4), 497–510.Google Scholar
da Costa, N. C. A., Béziau, J.-Y., & Bueno, O. (1995). Aspects of paraconsistent logic. Bulletin of the IGPL, 3(4), 597–614.CrossRefGoogle Scholar
D’Ottaviano, I. M. L. (1982). Sobre uma Teoria de Modelos Trivalente (On a Three-valued Model Theory, in Portuguese). Ph.D. Thesis, Brazil: IMECC, State University of Campinas.Google Scholar
D’Ottaviano, I. M. L. (1985a). The completeness and compactness of a three-valued first-order logic. Revista Colombiana de Matemáticas, XIX(1–2), 77–94.Google Scholar
D’Ottaviano, I. M. L. (1985b). The model extension theorems for J3-theories. In Di Prisco, C. A., editor. Methods in Mathematical Logic. Proceedings of the 6th Latin American Symposium on Mathematical Logic held in Caracas, Venezuela, August 1-6, 1983. Lecture Notes in Mathematics, Vol. 1130. Berlin: Springer-Verlag, pp. 157–173.Google Scholar
D’Ottaviano, I. M. L. (1987). Definability and quantifier elimination for J3-theories. Studia Logica, 46(1), 37–54.CrossRefGoogle Scholar
D’Ottaviano, I. M. L. & da Costa, N. C. A. (1970). Sur un problème de Jaśkowski (in French). Comptes Rendus de l’Académie de Sciences de Paris (A–B), 270, 1349–1353.Google Scholar
Ferguson, T. M. (2018). The Keisler-Shelah theorem for QmbC through semantical atomization. Logic Journal of the IGPL, to appear. First published online: November 29, 2018. Doi: 10.1093/jigpal/jzy067.Google Scholar
Fidel, M. M. (1978). An algebraic study of a propositional system of Nelson. In Arruda, A. I., da Costa, N. C. A., & Chuaqui, R., editors. Mathematical Logic. Proceedings of the First Brazilian Conference on Mathematical Logic, Campinas 1977. Lecture Notes in Pure and Applied Mathematics, Vol. 39. New York: Marcel Dekker, pp. 99–117.Google Scholar
Kalman, J. A. (1958). Lattices with involution. Transactions of the American Mathematical Society, 87, 485–491.CrossRefGoogle Scholar
Marcos, J. (2000). 8K Solutions and Semi-solutions to a Problem of da Costa. Unpublished draft.Google Scholar
Mendelson, E. (1987). Introduction to Mathematical Logic (third edition). New York: Chapman & Hall.CrossRefGoogle Scholar
Mikenberg, I., da Costa, N. C. A., & Chuaqui, R. (1986). Pragmatic truth and approximation to truth. The Journal of Symbolic Logic, 51(1), 201–221.CrossRefGoogle Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219–241.CrossRefGoogle Scholar
Priest, G. (2006). In Contradiction: A Study of the Transconsistent (second edition). Oxford: Oxford University Press.CrossRefGoogle Scholar
Sette, A. M. A. (1973). On the propositional calculus P 1 . Mathematica Japonicae, 18(13), 173–180.Google Scholar
Vakarelov, D. (1977). Notes on N-lattices and constructive logic with strong negation. Studia Logica, 36(1–2), 109–125.CrossRefGoogle Scholar
Wójcicki, R. (1984). Lectures on Propositional Calculi. Wroclaw, Poland: Ossolineum.Google Scholar