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Two Decision Procedures for da Costa’s \(C_n\) Logics Based on Restricted Nmatrix Semantics

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Abstract

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between \(\mathbf{mbCcl}\) and \(\mathbf{Cila}\). In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa’s logic \(C_1\)) but the whole hierarchy of da Costa’s calculi \(C_n\). This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each \(C_n\), which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa’s hierarchy of C-systems as a family of (non deterministically) \((n+2)\)-valued logics, where n is the number of “inconsistently true” truth-values and 2 is the number of “classical” or “consistent” truth-values, for every \(C_n\).

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References

  1. Alves, E. H., Lógica e inconsistência: um estudo dos cálculos\({\bf C}_{n}\), \(1\le n<\omega \)(Logic and inconsistency: a study of the calculi\({{\bf C}}_{n}\), \(1\le n<\omega \), in Portuguese), Ph.D. thesis, Universidade de São Paulo, São Paulo, Brazil, 1976.

  2. Avron, A., Non-deterministic matrices and modular semantics of rules, in J. Y. Beziau, (ed.), Logica Universalis, Birkhäuser, Basel, 2005, pp. 149–167.

    Chapter  Google Scholar 

  3. Avron, A., Non-deterministic semantics for paraconsistent C-systems, in L. Godo, (ed.), Proceedings of the VIII European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2005), vol. 3571 of Lecture Notes in Computer Science, Springer, Berlin, 2005, pp. 625–637.

  4. Avron, A., Non-deterministic semantics for logics with a consistency operator, International Journal of Approximate Reasoning 45(2):271–287, 2007.

    Article  Google Scholar 

  5. Avron, A., Paraconsistency and the need for infinite semantics, Soft Computing 23(7):2167–2175, 2019.

    Article  Google Scholar 

  6. Avron, A., O. Arieli, and A. Zamansky, Theory of Effective Propositional Paraconsistent Logics, vol. 75 of Studies in Logic (Mathematical Logic and Foundations), College Publications, 2018.

  7. Avron, A., and B. Konikowska, Multi-valued calculi for logics based on non-determinism, Logic Journal of the IGPL 13(4):365–387, 2005.

    Article  Google Scholar 

  8. Avron, A., and I. Lev, Canonical propositional Gentzen-type systems, in R. Goré, A. Leitsch, and T. Nipkow, (eds.), Proceedings of the First International Joint Conference on Automated Reasoning (IJCAR ’01), vol. 2083 of Lecture Notes in Artificial Intelligence, Springer-Verlag, London, 2001, pp. 529–544.

  9. Avron, A., and I. Lev, Non-deterministic multi-valued structures, Journal of Logic and Computation 15(3):241–261, 2005.

    Article  Google Scholar 

  10. Baaz, M., O. Lahav, and A. Zamansky, A finite-valued semantics for canonical labelled calculi, Journal of Automated Reasoning 51(4):401–430, 2013.

    Article  Google Scholar 

  11. Caleiro, C., and S. Marcelino, Analytic calculi for monadic PNmatrices, in R. Iemhoff, M. Moortgat, and R. de Queiroz, (eds.), Logic, Language, Information, and Computation, vol. 11541 of Lecture Notes in Computer Science, Springer, 2019, pp. 84–98.

  12. Carnielli, W. A., Systematization of finite many-valued logics through the method of tableaux, The Journal of Symbolic Logic 52(2):473–493, 1987.

    Article  Google Scholar 

  13. Carnielli, W. A., and M. E. Coniglio, Paraconsistent logic: Consistency, Contradiction and Negation, vol. 40 of Logic, Epistemology, and the Unity of Science, Springer, 2016.

  14. Carnielli, W. A., M. E. Coniglio, and J. Marcos, Logics of formal inconsistency, in D. M. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 14, Springer, 2007, pp. 1–93.

  15. Carnielli, W. A., and J. Marcos, A taxonomy of C-systems, in W. A. Carnielli, M. E. Coniglio, and I. M. D’Ottaviano, (eds.), Paraconsistency: The Logical Way to the Inconsistent, vol. 228 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 2002, pp. 1–94.

  16. Coniglio, M. E., L. Fariñas del Cerro, and N. M. Peron, Finite non-deterministic semantics for some modal systems, Journal of Applied Non-Classical Logic 25(1):20–45, 2015.

  17. Coniglio, M. E., L. Fariñas del Cerro, and N. M. Peron, Errata and addenda to Finite non-deterministic semantics for some modal systems, Journal of Applied Non-Classical Logic 26(4):336–345, 2016.

  18. Coniglio, M. E., L. Fariñas del Cerro, and N. M. Peron, Tableau systems for some Ivlev-like (quantified) modal logics, to appear, 2021.

  19. Coniglio, M. E., and A. Figallo Orellano, A model-theoretic analysis of Fidel-structures for mbC, in C. Baskent, and T. Ferguson, (eds.), Graham Priest on Dialetheism and Paraconsistency, vol. 14 of Outstanding Contributions to Logic, Springer, 2020, pp. 189–216.

  20. Coniglio, M. E., and G. V. Toledo, A simple decision procedure for da Costa’s \(C_n\) logics by Restricted Nmatrix semantics, arXiv:2011.10151 [math.LO], 2020.

  21. da Costa, N. C. A., Sistemas formais inconsistentes (Inconsistent Formal Systems, in Portuguese), Universidade do Paraná, Curitiba, 1963 (republished by Editora UFPR, Brazil 1993).

  22. da Costa, N. C. A., and E. H. Alves, A semantical analysis of the calculi \({\mathbf{C}}_{n}\), Notre Dame Journal of Formal Logic 18(4):621–630, 1977.

    Google Scholar 

  23. da Costa, N. C. A., and M. Guillaume, Négations composées et loi de Peirce dans les systèmes \({\bf C}_{n}\), Portugaliae Mathematicae 24: 201–210, 1965.

    Google Scholar 

  24. Dugundji, J., Note on a property of matrices for Lewis and Langford’s calculi of propositions, The Journal of Symbolic Logic 5(4):150–151, 1940.

  25. Fidel, M. M., The decidability of the calculi \({C}_n\), Reports on Mathematical Logic 8:31–40, 1977.

    Google Scholar 

  26. Gödel, K., Zum intuitionistischen aussagenkalkül, Anzeiger der AkademiederWissenschaften in Wien. Mathematisch-Naturwissenschaftliche Klasse 69:65–66, 1932 (translated as: On the intuitionistic propositional calculus, in: S. Feferman, J.W. Jr. Dawson, S.C. Kleene, G. Moore, R. Solovay, and J. Van Heijenoort, (eds.), Kurt Gödel, Collected Works: Publications 1929–1936, Oxford University Press, New York, 1986, pp. 222–225).

  27. Grätz, L., Analytic tableaux for non-deterministic semantics, in A. Das, and S. Negri, (eds.), Automated Reasoning with Analytic Tableaux and Related Methods, vol. 12842 of Lecture Notes in Artificial Intelligence, Springer International Publishing, 2021, pp. 38–55.

  28. Ivlev, Ju. V., Tablitznoe postrojenie propozicionalnoj modalnoj logiki (Truth-tables for systems of propositional modal logic, in Russian), Vest. Mosk. Univ., Seria Filosofia, 6 1973.

  29. Ivlev, Ju. V., A semantics for modal calculi, Bulletin of the Section of Logic 17(3/4):114–121, 1988.

    Google Scholar 

  30. Jaśkowski, S., Rachunek zdan dla systemów dedukcyjnych sprzecznych, Studia Societatis Scientiarum Torunensis, 1(5):55–77, 1948 (translated as: Propositional calculus for contradictory deductive systems, Studia Logica 24:143–157, 1969).

  31. Jaśkowski, S., O koniunkcji dyskusyjnej w rachunku zdan dla systemów dedukcyjnych sprzecznych, Studia Societatis Scientiarum Torunensis 8:171–172, 1949 (translated as: On the discussive conjunction in the propositional calculus for inconsistent deductive systems, Logic and Logical Philosophy 7:57–59, 1999).

  32. Kearns, J. T., Modal semantics without possible worlds, The Journal of Symbolic Logic 46(1):77–86, 1981.

    Article  Google Scholar 

  33. Loparić, A., and E. H. Alves, The semantics of the systems \({\bf C}_{n}\) of da Costa, in A. I. Arruda, N. C. A. da Costa, and A. M. A. Sette, (eds.), Proceedings of the Third Brazilian Conference on Mathematical Logic, Sociedade Brasileira de Lógica, Recife, Brazil, 1980, pp. 161–172.

  34. Omori, H., and D. Skurt, More modal semantics without possible worlds, IfCoLog Journal of Logics and their Applications 3(5):815–846, 2016.

    Google Scholar 

  35. Omori, H., and D. Skurt, A semantics for a failed axiomatization of \(K\), in N. Olivietti, R. Verbrugge, S. Negri, and G. Sandu, (eds.), Advances in Modal Logic, vol. 13, College Publications, 2020, pp. 481–501.

  36. Pawlowski, P, Tree-like proof systems for finitely-many valued non-deterministic consequence relations, Logica Universalis 14(4):407–420, 2020.

    Article  Google Scholar 

  37. Pawlowski, P., and R. Urbaniak, Many-valued logic of informal provability: a non-deterministic strategy, Review of Symbolic Logic 11(2):207–223, 2018.

    Article  Google Scholar 

  38. Piochi, B., Matrici adequate per calcoli generali predicativi, Bolletino della Unione Matematica Italiana 15A:66–76, 1978.

    Google Scholar 

  39. Piochi, B., Logical matrices and non-structural consequence operators, Studia Logica 42(1):33–42, 1983.

    Article  Google Scholar 

  40. Rescher, N., Quasi-truth-functional systems of propositional logic, The Journal of Symbolic Logic 27(1):1–10, 1962.

    Article  Google Scholar 

  41. Smullyan, R. M., First-Order Logic, Dover Publications, Mineola, N.Y. USA, 1995 (corrected republication of the Springer-Verlag, New York, 1968 edition).

  42. Wójcicki, R., Some remarks on the consequence operation in sentential logics, Fundamenta Mathematicae 68(3):269–279, 1970.

    Article  Google Scholar 

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Acknowledgements

We would like to thank the anonymous referees by their careful reading and useful suggestions which helped us to improve the overall quality of this paper. This paper is a corrected and expanded version of the preliminary draft [20]. The first author acknowledges support from the National Council for Scientific and Technological Development (CNPq), Brazil, under research Grant 306530/2019-8. The second author was supported by a doctoral scholarship from CAPES, Brazil.

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Authors and Affiliations

  1. Institute of Philosophy and the Humanities - IFCH, University of Campinas - Unicamp, R. Cora Coralina, 100, Campinas, SP, Brazil

    Marcelo E. Coniglio & Guilherme V. Toledo

  2. Centre for Logic, Epistemology and The History of Science - CLE, University of Campinas - Unicamp, R. Sérgio Buarque de Holanda, 251, Campinas, SP, Brazil

    Marcelo E. Coniglio

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  1. Marcelo E. Coniglio
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Correspondence to Marcelo E. Coniglio.

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Dedicated to Newton C.A. da Costa, a permanent source of inspiration.

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Coniglio, M.E., Toledo, G.V. Two Decision Procedures for da Costa’s \(C_n\) Logics Based on Restricted Nmatrix Semantics. Stud Logica 110, 601–642 (2022). https://doi.org/10.1007/s11225-021-09972-z

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