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AN ABSTRACT ALGEBRAIC LOGIC STUDY OF DA COSTA’S LOGIC ${\mathscr {C}}_1$ AND SOME OF ITS PARACONSISTENT EXTENSIONS

Part of: General Algebraic logic General logic

Published online by Cambridge University Press:  04 October 2022

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HUGO ALBUQUERQUE
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INDEPENDENT RESEARCHER E-mail: hugo.albuquerque@ua.pt
CARLOS CALEIRO
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SQIG, INSTITUTO DE TELECOMUNICAÇÕES LISBON, PORTUGAL and DEPARTAMENTO DE MATEMÁTICA, INSTITUTO SUPERIOR TÉCNICO UNIVERSIDADE DE LISBOA LISBON, PORTUGAL E-mail: ccal@math.tecnico.ulisboa.pt
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Abstract

Two famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$ . On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as , , or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$ , with , , and .

Keywords

da Costa’s logic C1paraconsistent logicsprotoalgebraic logicsabstract algebraic logic

MSC classification

Primary: 00-01: Instructional exposition (textbooks, tutorial papers, etc.) 03G27: Abstract algebraic logic 03B53: Paraconsistent logics
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 4 , December 2022 , pp. 477 - 528
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© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Albuquerque, H., Font, J. M., and Jansana, R., The strong version of a sentential logic . Studia Logica , vol. 105 (2017), no. 4, pp. 703–760.CrossRefGoogle Scholar
Arruda, A. I., Aspects of the historical development of paraconsistent logic , Paraconsistent Logic: Essays on the Inconsistent (G. Priest, R. Routley and J. Norman, editors), Philosophia, 1989, pp. 99–130.CrossRefGoogle Scholar
Arruda, A. I. and da Costa, N. C. A., Sur Une hiérarchie de systèmes formels . Comptes rendus de l’Académie des Sciences Paris , vol. 259 (1964), pp. 2943–2945.Google Scholar
Bergman, C., Universal Algebra: Fundamentals and Selected Topics , Pure and Applied Mathematics, vol. 301, CRC Press, Boca Raton, FL, 2012.Google Scholar
Birkhoff, G., Subdirect unions in universal algebra . Bulletin of the American Mathematical Society , vol. 50 (1944), pp. 764–768.CrossRefGoogle Scholar
Blok, W. J. and Pigozzi, D., Algebraizable logics . Memoirs of the American Mathematical Society , vol. 77 (1989), no. 396.CrossRefGoogle Scholar
Blok, W. J. and Raftery, J. G., Assertionally equivalent quasivarieties . International Journal of Algebra and Computation , vol. 18 (2008), no. 4, pp. 589–681.CrossRefGoogle Scholar
Blok, W. J. and Rebagliato, J., Algebraic semantics for deductive systems . Studia Logica , vol. 74 (2003), nos. 1–2, pp. 153–180.CrossRefGoogle Scholar
Bueno-Soler, J. and Carnielli, W., Possible-translations algebraization for paraconsistent logics . Bulletin of the Section of Logic , vol. 34 (2005), no. 2, pp. 77–92.Google Scholar
Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra , Graduate Texts in Mathematics, vol. 78, Springer, New York, 1981.CrossRefGoogle Scholar
Caicedo, X., The subdirect decomposition theorem for classes of structures closed under direct limits . Australian Mathematical Society , vol. 30 (1980/81), no. 2, pp. 171–179.CrossRefGoogle Scholar
Caleiro, C. and Gonçalves, R., Behavioral algebraization of da Costa’s $C$ -systems . Journal of Applied Non-Classical Logics , vol. 19 (2009), no. 2, pp. 127–148.CrossRefGoogle Scholar
Caleiro, C., Gonçalves, R., and Martins, M., Behavioral algebraization of logics . Studia Logica , vol. 91 (2009), no. 1, pp. 63–111.CrossRefGoogle Scholar
Carnielli, W. A. and de Alcantara, L. P., Paraconsistent algebras . Studia Logica , vol. 43 (1984), nos. 1–2, pp. 79–88.CrossRefGoogle Scholar
Carnielli, W. A. and Marcos, J., Limits for paraconsistent calculi . Notre Dame Journal of Formal Logic , vol. 40 (1999), no. 3, pp. 375–390.CrossRefGoogle Scholar
Carnielli, W. A. and Marcos, J., A taxonomy of C-systems , Paraconsistency (São Sebastião, 2000) , Lecture Notes in Pure and Applied Mathematics, vol. 228, Dekker, New York, 2002, pp. 1–94.Google Scholar
Carnielli, W. A., Coniglio, M. E., and Marcos, J., Logics of formal inconsistency , Handbook of Philosophical Logic , vol. 14, second ed. (D. M. Gabbay and F. Guenthner, editors), Springer, Dordrecht, 2007, pp. 1–93.Google Scholar
da Costa, N. C. A., Calculs propositionnels pour les systèmes formels inconsistants . Comptes Rendus de l’Académie des Sciences Paris , vol. 257 (1963), pp. 3790–3792.Google Scholar
da Costa, N. C. A., Opérations non monotones dans les treillis . Comptes Rendus de l’Académie des Sciences Paris Séries A–B , vol. 263 (1966), pp. A429–A432.Google Scholar
da Costa, N. C. A., Filtres et idéaux d’une algèbre ${C}_n$ . Comptes Rendus de l’Académie des Sciences Paris Séries A–B , vol. 264 (1967), pp. A549–A552.Google Scholar
da Costa, N. C. A., On the theory of inconsistent formal systems . Notre Dame Journal of Formal Logic , vol. 15 (1974), pp. 497–510.CrossRefGoogle Scholar
da Costa, N. C. A. and Alves, E. H., Une sémantique pour le calcul ${C}_1$ . Comptes Rendus de l’Académie des Sciences Paris Séries A–B , vol. 283 (1976), no. 10, pp. A729–A731.Google Scholar
da Costa, N. C. A. and Alves, E. H., A semantical analysis of the calculi ${C}_n$ . Notre Dame Journal of Formal Logic , vol. 18 (1977), no. 4, pp. 621–630.Google Scholar
da Costa, N. C. A., Béziau, J.-Y., and Bueno, O. A. S., Aspects of paraconsistent logic . Bulletin of the IGPL , vol. 3 (1995), no. 4, pp. 597–614. Workshop on Logic, Language, Information and Computation’94 (Recife, 1994).CrossRefGoogle Scholar
da Costa, N. C. A. and Guillaume, M., Sur les calculs ${C}_n$ . Anais da Academia Brasileira de Ciências , vol. 36 (1964), pp. 379–382.Google Scholar
da Costa, N. C. A., Krause, D., and Bueno, O., Paraconsistent logics and paraconsistency: Technical and philosophical developments. Available at ftp://www.cle.unicamp.br/pub/e-prints/vol.4,n.3,2004.pdf.Google Scholar
da Costa, N. C. A. and Sette, A. M., Les algèbres ${C}_{\omega }$ . Comptes Rendus de l’Académie des Sciences Paris Séries A–B , vol. 268 (1969), pp. A1011–A1014.Google Scholar
Czelakowski, J., Equivalential logics. I . Studia Logica , vol. 40 (1981), no. 3, pp. 227–236.CrossRefGoogle Scholar
Czelakowski, J. and Jansana, R., Weakly algebraizable logics . The Journal of Symbolic Logic , vol. 65 (2000), pp. 641–668.Google Scholar
Fidel, M. M., The decidability of the calculi ${C}_n$ . Reports on Mathematical Logic , vol. 8 (1977), pp. 31–40.Google Scholar
Font, J. M., The simplest protoalgebraic logic . Mathematical Logic Quarterly , vol. 59 (2013), no. 6, pp. 435–451.CrossRefGoogle Scholar
Font, J. M., Abstract Algebraic Logic: An Introductory Textbook , Studies in Logic, vol. 60, College Publications, London, 2016.Google Scholar
Font, J. M. and Jansana, R., A General Algebraic Semantics for Sentential Logics , second revised edition, Lecture Notes in Logic, vol. 7, Association for Symbolic Logic, 2009. Available at http://projecteuclid.org/euclid.lnl/1235416965. First edition published by Springer, Berlin, 1996.Google Scholar
Gabbay, D. M. and Guenthner, F., Handbook of Philosophical Logic , vol. 6 , second ed., Springer, Dordrecht, 2007.Google Scholar
Gorbunov, V. A., Algebraic Theory of Quasivarieties , Siberian School of Algebra and Logic, Consultants Bureau, New York, 1998. Translated from the Russian.Google Scholar
Herrmann, B., Equivalential and algebraizable logics . Studia Logica , vol. 57 (1996), pp. 419–436.CrossRefGoogle Scholar
Jansana, R., Abstract algebraic logic. Lecture notes, 2014.Google Scholar
Lewin, R. A., Mikenberg, I. F., and Schwarze, M. G., Algebraization of paraconsistent logic ${P}^1$ . Journal of Non-Classical Logic , vol. 7 (1990), nos. 1–2, pp. 79–88.Google Scholar
Lewin, R. A., Mikenberg, I. F., and Schwarze, M. G., ${C}_1$ is not algebraizable . Notre Dame Journal of Formal Logic , vol. 32 (1991), no. 4, pp. 609–611 (1992).Google Scholar
Lewin, R. A., Mikenberg, I. F., and Schwarze, M. G., P1 algebras . Studia Logica , vol. 53 (1994), no. 1, pp. 21–28.CrossRefGoogle Scholar
Mal’cev, A. I., Subdirect products of models . Akademii Nauk SSSR (N.S.) , vol. 109 (1956), pp. 264–266.Google Scholar
Marcos, J., Logics of formal inconsistency , Ph.D. thesis, Universidade Técnica de Lisboa, 2005. Available at https://www.math.tecnico.ulisboa.pt/~jmarcos/Thesis/.Google Scholar
Moraschini, T., On equational completeness theorems , 2020. Available at https://moraschini.github.io/files/submitted/equational.pdf.CrossRefGoogle Scholar
Mortensen, C., Every quotient algebra for ${C}_1$ is trivial . Notre Dame Journal of Formal Logic , vol. 21 (1980), no. 4, pp. 694–700.Google Scholar
Mortensen, C., Paraconsistency and 𝓒1 , Paraconsistent Logic: Essays on the Inconsistent (G. Priest, R. Routley and J. Norman, editors), Philosophi, 1989, pp. 289–305.CrossRefGoogle Scholar
Priest, G., Paraconsistency and dialetheism , Handbook of the History of Logic , vol. 8 (D. M. Gabbay and J. Woods, editors), Elsevier, North-Holland, Amsterdam, 2007, pp. 129–204.Google Scholar
Priest, G., Paraconsistent logic , Handbook of Philosophical Logic , vol. 6, second ed. (D. M. Gabbay and F. Guenthner, editors), Springer, Dordrecht, 2007, pp. 287–393.Google Scholar
Priest, G., Routley, R., and Norman, J., Paraconsistent Logic: Essays on the Inconsistent , Philosophia, 1989.Google Scholar
Priest, G., Tanaka, K., and Weber, Z., Paraconsistent logic, Stanford Encyclopedia of Philosophy . Available at https://plato.stanford.edu/entries/logic-paraconsistent/ (accessed 20 July, 2022).Google Scholar
Pynko, A. P., Algebraic study of Sette’s maximal paraconsistent logic . Studia Logica , vol. 54 (1995), no. 1, pp. 89–128.CrossRefGoogle Scholar
Raftery, J. G., The equational definability of truth predicates . Reports on Mathematical Logic , vol. 41 (2006), pp. 95–149.Google Scholar
Seoane, J. and de Alcantara, L. P., On da Costa algebras . Journal Non-Classical Logic , vol. 8 (1991), no. 2, pp. 41–66.Google Scholar
Sette, A. M., On the propositional calculus ${P}^1$ . Mathematica Japonica , vol. 18 (1973), pp. 173–180.Google Scholar
Urbas, I., Paraconsistency and the $C$ -systems of da Costa . Notre Dame Journal of Formal Logic , vol. 30 (1989), no. 4, pp. 583–597.CrossRefGoogle Scholar
Weber, Z., Paraconsistent logic, Internet Encyclopedia of Philosophy . Available at https://iep.utm.edu/para-log/ (accessed 20 July, 2022).Google Scholar
Wójcicki, R., Theory of Logical Calculi: Basic Theory of Consequence Operations , Synthese Library, vol. 199, Kluwer Academic, Dordrecht, 1988.Google Scholar