Abstract
In order to develop efficient tools for automated reasoning with inconsistency (theorem provers), eventually making Logics of Formal inconsistency (LFI) a more appealing formalism for reasoning under uncertainty, it is important to develop the proof theory of the first-order versions of such LFIs. Here, we intend to make a first step in this direction. On the other hand, the logic Ciore was developed to provide new logical systems in the study of inconsistent databases from the point of view of LFIs. An interesting fact about Ciore is that it has a strong consistency operator, that is, a consistency operator which (forward/backward) propagates inconsistency. Also, it turns out to be an algebraizable logic (in the sense of Blok and Pigozzi) that can be characterized by means of a 3-valued logical matrix. Recently, a first-order version of Ciore, namely QCiore, was defined preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. Besides, some important model-theoretic results were obtained for this logic. In this paper we study some proof–theoretic aspects of both Ciore and QCiore respectively. In first place, we introduce a two-sided sequent system for Ciore. Later, we prove that this system enjoys the cut-elimination property and apply it to derive some interesting properties. Later, we extend the above-mentioned system to first-order languages and prove completeness and cut-elimination property using the well-known Shütte’s technique.
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References
Avron, A., Non-deterministic semantics for logics with a consistency operator, Journal of Approximate Reasoning 45: 271–287, 2007.
Avron, A., J. Ben-Naim, and B. Konikowska, Cut-free ordinary sequent calculi for logics having generalized finite–valued semantics, Logica Universalis 1: 41–69, 2006.
Avron, A., and B. Konikowska, Multi-valued Calculi for logics based on non-determinism, in Proceedings COS’04 (Challenge of Semantics Workshop), Vienna 2004; also in: Logic Journal of the IGPL 13(4): 365–387, 2005.
Avron, A., B. Konikowska, and A. Zamansky, Modular construction of cut-free sequent Calculi for paraconsistent logics, in Proceedings of the 27th Annual ACM/IEEE Symposium on Logic in Computer Science, 2012, pp. 85–94.
Cantú, L., Sobre la lógica que preserva grados de verdad asociada a las álgebras de Stone involutivas. Masters dissertation, Universidad Nacional del Sur (Bahía Blanca, Argentina), 2019.
Cantú, L., and M. Figallo, On the logic that preserves degrees of truth associated to involutive Stone algebras, Logic Journal of the IGPL 28(5): 1000–1020, 2020.
Cantú, L., and M. Figallo, Cut-free sequent-style systems for a logic associated to involutive Stone algebras, Journal of Logic and Computation 33(7):1684–1710, 2023. https://doi.org/10.1093/logcom/exac061
Carnielli, W., and M. E. Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, vol. 40 of Logic, Epistemology, and the Unity of Science Series, Springer, 2016.
Carnielli, W. A., M. E. Coniglio, and J. Marcos, Logics of formal inconsistency, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 14, Springer, 2007, pp. 1–93.
Carnielli, W. A., and J. Marcos, A taxonomy of C-systems, in W. A. Carnielli, M. E. Coniglio, and I. M. L. D’Ottaviano, (eds.), Paraconsistency—The Logical Way to the Inconsistent, vol. 228 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 2002, pp. 1–94.
Carnielli, W. A., J. Marcos, and S. de Amo, Formal inconsistency and evolutionary databases, Logic and Logical Philosophy 8: 115–152, 2000.
Coniglio, M. E., G. T. Gomez-Pereira, and M. Figallo, Some model–theoretic results on the 3-valued paraconsistent first-order logic QCiore, The Review of Symbolic Logic 4(1): 187–224, 2021. https://doi.org/10.1017/S1755020319000595
da Costa, N. C. A., Inconsistent Formal Systems (in Portuguese), Habilitation Thesis, 1963. Republished by Editora UFPR, Curitiba, 1993.
da Costa, N. C. A., 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.
da Costa, N. C. A., Calculs propositionnel pour les systèmes formels inconsistants, Comptes Rendus de l’Académie de Sciences de Paris, série A 257: 3790–3792, 1963.
da Costa, N. C. A., J.-Y. Béziau, and O. Bueno, Aspects of paraconsistent logic, Bulletin of the IGPL 3(4): 597–614, 1995.
Figallo, M., Cut-free sequent calculus and natural deduction for the tetravalent modal logic, Studia Logica 109: 1347–1373, 2021. https://doi.org/10.1007/s11225-021-09944-3
Mikenberg, I., N. C. A. da Costa, and R. Chuaqui, Pragmatic truth and approximation to truth, The Journal of Symbolic Logic 51(1): 201–221, 1986.
Takeuti, G., Proof Theory (2nd edn.), Dover Publications, Mineola, New York, 2013 [1975]
Acknowledgements
V. A. Pistone is grateful to CONICET for providing financial support for this research. M. Figallo was partially supported by the Visiting Researcher Award program funded by FAPESP grant 2022/03862-2.
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Arce Pistone, V., Figallo, M. Proof-Theoretic Aspects of Paraconsistency with Strong Consistency Operator. Stud Logica (2024). https://doi.org/10.1007/s11225-023-10089-8
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DOI: https://doi.org/10.1007/s11225-023-10089-8