Lógica trivalorada e paraconsistência em Peirce
DOI:
https://doi.org/10.23925/2316-5278.2025v26i1:e57507Palavras-chave:
Charles S. Peirce, Lógica, Lógicas não-clássicas, Lógicas trivaloradas, Paraconsistência, SequentesResumo
Peirce é hoje reconhecido como um dos pioneiros da lógica matemática e algébrica, mas seu trabalho original em lógicas não-clássicas ainda recebe escassa atenção fora do círculo estreito de especialistas peircianos. Esse é o caso do cálculo proposicional trivalorado que Peirce registrou em seu “Logic Notebook”, mais de uma década antes do surgimento das lógicas multivaloradas. A lógica triádica, como Peirce a chamava, foi formalizada por Turquette no final dos anos 1960. Turquette apresentou uma interpretação axiomática das tabelas trivaloradas em uma série de artigos que se tornaram referência nesse estudo. Recentemente, propusemos uma nova abordagem, enfatizando um fragmento não-explosivo da lógica triádica. Este artigo objetiva ampliar a pesquisa nos seguintes pontos: (i) uma análise crítica dos trabalhos de Turquette, incluindo a discussão do método Rosser-Turquette de axiomatização; e (ii) reconstrução do fragmento da lógica triádica em um sistema baseado na implicação material de Sobociński, formalizado em cálculo de sequentes. Concluímos que a matriz trivalorada de Peirce induz uma lógica paraconsistente, relevante e subestrutural, com potencial investigativo para as pesquisas contemporâneas em lógicas não-clássicas.Metrics
Referências
ARIELI, O.; AVRON, A. Three-valued paraconsistent propositional logics. In: BEZIAU, J-Y et al. (eds.). New directions in paraconsistent logic. New Delhi, India: Springer, 2015. p. 91-129.
AVRON, A. Natural 3-valued logics: characterization and proof theory. The Journal of Symbolic Logic, v. 56, n. 1, p. 276-294, Mar., 1991. https://doi.org/10.2307/2274919.
AVRON, A. Classical Gentzen-type methods in propositional many-valued logics. In: FITTING, M.; ORŁOWSKA, E. (eds). Beyond two: theory and applications of multiple-valued logic. Springer-Verlag: Heidelberg, 2003. p. 117-155.
BELIKOV, A. Peirce’s triadic logic and its (overlooked) connexive expansion. Logic and Logical Philosophy, 2021 (online first articles). Available at: https://apcz.umk.pl/czasopisma/index.php/LLP/article/view/LLP.2021.007. Accessed 16 June 2021.
BELNAP, N. D. Conditional assertion and restricted quantification. Noûs, v. 4, n. 1, p. 1-12, 1970. https://doi.org/10.2307/2214285.
BOCHVAR, D. A; BERGMANN, M On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. History and Philosophy of Logic, v. 2, n. 1-2, p. 87-112, 1981. https://doi.org/10.1080/01445348108837023.
BRADY, G. From Peirce to Skolem: a neglected chapter in the History of Logic. Amsterdam: Elsevier, 2000.
CARNIELLI, W., MARCOS, J., DE AMO, S. Formal inconsistency and evolutionary databases. Logic and Logical Philosophy, v. 8, n. 8, p. 115-152, 2000. https://doi.org/10.12775/LLP.2000.008.
CARNIELLI, W., MARCOS, J. A taxonomy of C-systems. In: CARNIELLI, W., CONIGLIO, M. E., D’OTTAVIANO, I. M. L. (eds.). Paraconsistency: the logical way to the inconsistent. Boca Raton: CRC Press, 2002. p. 01-94.
CARNIELLI, W., CONIGLIO, M. E. Paraconsistent logic: consistent, contradiction and negation. Dordrecht: Springer, 2016.
CARUS, P. The nature of logical and mathematical thought. The Monist, v. 20, n. 1, p. 33-75, Jan. 1910. https://doi.org/10.5840/monist191020120.
CHANG, C. C. Algebraic analysis of many valued logics. Transactions of the American Mathematical Society, v. 88, n. 2, p. 467-490, Jul. 1958. https://doi.org/10.1090/S0002-9947-1958-0094302-9.
COOPER, W. S. The propositional logic of ordinary discourse. Inquiry: An Interdisciplinary Journal of Philosophy, v. 11, n. 1-4, p. 295-320, 1968. https://doi.org/10.1080/00201746808601531.
COSTA, N. C. da. Sistemas formais inconsistentes. Curitiba: UFPR, 1993.
BELIKOV, A. Peirce’s triadic logic and its (overlooked) connexive expansion. Logic and Logical Philosophy, 2021 (online first articles). Available at: https://apcz.umk.pl/czasopisma/index.php/LLP/article/view/LLP.2021.007. Accessed 16 June 2021.
FISCH, M. e TURQUETTE, A. Peirce’s triadic logic. Transactions of the Charles S. Peirce Society, v. II, n. 2, p. 71-85, 1966.
KLEENE, S. On notation for ordinal numbers. Journal of Symbolic Logic, v. 3, n. 4, p.150-155, 1938. https://doi.org/10.2307/2267778.
KLEENE, S. Introduction to metamathematics. Amsterdam/ Oxford: North-Holland Publishing Company, 1971.
LANE, R. Peirce’s triadic logic revisited. Transactions of the Charles S. Peirce Society, v. 35, n. 2, p. 284-311, 1999.
LANE, R. Triadic logic. In: BERGMAN, M.; QUEIROZ, J. (eds.). The Commens Encyclopedia: The Digital Encyclopedia of Peirce Studies. New Edition, 2001. Available at: http://www.commens.org/encyclopedia/article/lane-robert-triadic-logic. Acessed: 12 February 2021.
ŁUKASIEWICZ, J. [1920]. On three-valued logic. In: BORKOWSKI, L. (ed.). Selected works. Amsterdam: North-Holland, 1970. p. 87-88.
MA, M.; PIETARINEN, A-V. Peirce’s calculi for classical propositional logic. The Review of Symbolic Logic, v. 13, n. 3, p. 509-540, 2020. https://doi.org/10.1017/S1755020318000187.
MALINOWSKI, G. Many-valued logics. Oxford: Oxford University Press, 1993.
MEYER, R. K.; PARKS, Z. Independent axioms for the implicational fragment of Sobociński’s three‐valued logic. Mathematical Logic Quarterly, v. 18 (19-20), p. 291-295, 1972. https://doi.org/10.1002/malq.19720181903.
ODLAND, B. C. Peirce’s triadic logic: continuity, modality, and L. Unpublished master’s thesis. University of Calgary: Calgary, AB, 2020. Available at: http://hdl.handle.net/1880/112238. Accessed 10 April 2021.
ODLAND, B. C. Peirce’s triadic logic: modality and continuity. Transactions of the Charles S. Peirce Society, v. 57, n. 2, p. 149-171.2021. https://doi.org/10.2979/trancharpeirsoc.57.2.01.
PARKS, R. Z. The mystery of Phi and Psi. Transactions of the Charles S. Peirce Society, v. 7, n. 3, p. 176–177, 1971.
PARKS, R. Z. A note on R-mingle and Sobociński’s three-valued logic. Notre Dame Journal of Formal Logic, v. 13, n. 2, p. 227-228, 1972. https://doi.org/10.1305/ndjfl/1093894720.
PEIRCE, C. S. On the algebra of logic: a contribution to the philosophy of notation. American Journal of Mathematics, v. 7, n. 2, p. 180-196, 1885. https://doi.org/10.2307/2369451.
PEIRCE, C. S. Collected Papers. 8 vols. HARTSHORNE, Charles; HEISS, Paul and BURKS, Arthur (eds.). Cambridge: Harvard University Press, 1931-1958.
PEIRCE, C. S. The new elements of mathematics. The Hague: Mouton Publishers, 1976.
PEIRCE, C. S. The essential Peirce: selected philosophical writings, v. 1 (1867-1893). HOUSER, N. and KLOESEL, C. (eds.). Indiana University Press: Bloomington and Indianapolis, 1992.
PEIRCE, C. S. The Essential Peirce: selected philosophical writings, v. 2 (1893-1913). The Peirce Edition Project (ed.). Indiana University Press: Bloomington and Indianapolis, 1998.
PEIRCE, C. S. Philosophy of mathematics: selected writings. MOORE, M. E. (ed.). Indiana University Press: Bloomington and Indianapolis, 2010.
PEIRCE, C. S. Peirce logic notebook, Charles Sanders Peirce Papers MS Am 1632 (339). Houghton Library, Harvard University, Cambridge, Mass. Available at: https://nrs.harvard.edu/urn-3:FHCL.Hough:3686182. Accessed 10 April 2021.
PEIRCE, C.S. Writings of Charles S. Peirce: a chronological edition, v. I-VIII. PEIRCE EDITION PROJECT (ed.). Indiana University Press: Bloomington, Indiana, 1982-2010.
PEIRCE, C. S. Descriptions of a notation for the logic of relatives, resulting from an amplification of Boole’s calculus of logic. In: MOORE, E, C. et al (ed.). Writings of Charles S. Peirce: a chronological edition, v. II Indiana University Press: Bloomington, Indiana, 1984.
PEIRCE, C. S. On the algebra of logic. In: KLOESEL, J. W. et al (ed.). Writings of Charles S. Peirce: a chronological edition, v. IV. Indiana University Press: Bloomington, Indiana, 1989.
PEIRCE, C. S. On the algebra of logic: a contribution to the philosophy of notation. In: KLOESEL, J. W. et al (ed.). Writings of Charles S. Peirce: a chronological edition, v. V. Indiana University Press: Bloomington, Indiana, 1993.
POST, E. Introduction to a general theory of elementary propositions. American Journal of Mathematics, v. 43, n. 3, p. 163-185, 1921. https://doi.org/10.2307/2370324.
RADZKI, M. On the Rosser-Turquette method of constructing axiom systems for many-valued propositional logics of Łukasiewicz. Journal of Applied Non-classical Logics, v. 27, n. 1-2, p. 27-32, 2017. https://doi.org/10.1080/11663081.2017.1311147.
RADZKI, M. Some problems concerning axiom systems for finitely many-valued propositional logics. In: DRABAREK, A. et al. (eds.). Interdisciplinary Investigations into the Lvov-Warsaw School. Palgrave Macmillan: Cham, 2019. p. 205-216.
RADZKI, M. On the methods of constructing Hilbert-type axiom systems for finite-valued propositional logics of Łukasiewicz. History and Philosophy of Logic, v. 43, n. 1, p. 70-79, 2022. https://doi.org/10.1080/01445340.2021.1899475.
RASIOWA, H. An algebraic approach to non-classical logics. Amsterdam: Polish Scientific Publishers, 1974.
RESCHER, N. Many-valued logic. New York: McGraw Hill, 1968.
ROBERTS, D. The existential graphs of Charles S. Peirce. The Hague: Mouton, 2009.
RODRIGUES, C. Squaring the unknown: the generalization of logic according to G. Boole, A. De Morgan, and C. S. Peirce. South American Journal of Logic, v. 3, n. 2, p. 415-481, 2017.
RODRIGUES, C. About a most subtle and disputed question: whether C. S. Peirce invented truth-tables, and why it does not matter. Version 4. 8/11/2021.
ROSE, A. An alternative formalization of Sobociński's three-valued implicational propositional calculus. Mathematical Logic Quarterly, v. 2, p. 166-172, 1956. https://doi.org/10.1002/malq.19560021003.
ROSSER, J. B.; TURQUETTE, A. R. Axiom schemes for m-valued propositional calculi. The Journal of Symbolic Logic, v. 10, n. 3, 1945, p. 61-82. https://doi.org/10.2307/2267026.
ROSSER, J. B.; TURQUETTE, A. R. Many-valued logics. Amsterdam: North-Holland Publishing Company, 1952.
SALATIEL, J.R. On a new approach to Peirce’s three-value propositional logic. Manuscrito: Revista Internacional de Filosofia, Campinas, SP, v. 45, n. 4, p. 79-106, 2023. Accessed 28 jun. 2024. https://doi.org/10.1590/0100-6045.2022.v45n4.js.
SALATIEL J.R. Tableau method of proof for Peirce’s three-valued propositional logic.
Unisinos Journal of Philosophy, v. 23, n. 1, p. 1-10, 2022. Accessed 28 jun. 2024. https://doi.org/10.4013/fsu.2022.231.05.
SOBOCIŃSKI, B. Axiomatization of a partial system of three-valued calculus of propositions. The Journal of Computing Systems, vo. I, p. 23-55, 1952.
SOBOCIŃSKI, B. Review: J. B. Rosser, A. R. Turquette, Many-Valued Logics. Journal of Symbolic Logic, v. 20, n. 1, p. 45-50, 1955. https://doi.org/10.2307/2268043.
SOBOCIŃSKI, B. A note concerning the many-valued propositional calculi. Notre Dame Journal of Formal Logic, v. 2, n. 2, p. 127-128, 1961. https://doi.org/10.1305/ndjfl/1093956835.
TARSKI, A. Introduction to logic and the methodology of deductive sciences. Oxford: Oxford University Press, 1994.
TURQUETTE, A. R. Review: Bolesław Sobociński, A note concerning the many-valued propositional calculi. Journal of Symbolic Logic, v. 31, n. 1, p. 117. 1966. https://doi.org/10.2307/2270643.
TURQUETTE, A. R. Peirce’s Phi and Psi operators for triadic logic. Transactions of the Charles S. Peirce Society, v. 3, n, 2, p. 66-73, 1967.
TURQUETTE, A. R. Peirce’s complete systems of triadic logic. Transactions of the Charles S. Peirce Society, v. 5, n. 4, p. 199-210, 1969.
TURQUETTE, A. R. Dualism and trimorphism in Peirce’s triadic logic. Transactions of the Charles S. Peirce Society, v. 8, n. 3, p. 131-140, 1972.
TURQUETTE, A. R. Implication for Peirce’s triadic logic. Proceedings of the XVth World Congress of Philosophy, v. 3, p. 399-401, 1974. https://doi.org/10.5840/wcp151974387.
TURQUETTE, A. R. Minimal axioms for Peirce’s triadic logic. Zeitschrift für mathe- matische Loßik und Grundlagen der Mathematik, v. 22, p. 169-176, 1976. https://doi.org/10.1002/malq.19760220123.
TURQUETTE, A. R. Alternative axioms for Peirce’s triadic logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, v. 24, p. 443-444, 1978. https://doi.org/10.1002/malq.19780242510.
TURQUETTE, A. R. Quantification for Peirce’s preferred system of triadic logic. Studia Logica, v. 40, p. 373-382, 1981. https://doi.org/10.1007/BF00401655.
TURQUETTE, A. R. Defining Peirce’s verum. Philosophie et Culture: Actes du XVIIe congrès mondial de philosophie, v. 2, p. 842-845, 1988. https://doi.org/10.5840/wcp1719882694.
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