On duality of logical systems: paralogics
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02.07.2025
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Abstract
The paper is devoted to the issues of the duality of logical systems. It is customary to consider paraconsistent and paracomplete logics as dual. Since systems with paraconsistent and paracomplete properties are found in the literature under various names, the issue of terminology is investigated separately. Based on well-known research, the phenomenon of the duality of paraconsistent and paracomplete systems is considered in various aspects: from the point of view of the deductive apparatus, the theory of logical matrices, algebraic
structures, as well as a relatively intuitive, meaningful interpretation of paralogics.
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How to Cite
Tomova N. E. On duality of logical systems: paralogics // Logicheskie Issledovaniya / Logical Investigations. 2025. VOL. 31. № 1. C. 47-73.
Issue
Section
Non-classical logics
Copyright (c) 2025 Наталья Евгеньевна Томова
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
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Ciuni, 2015 – Ciuni R. Conjunction in paraconsistent weak Kleene logic // Logica Yearbook 2014 / Ed. by P. Arazim and M. Danc´ak. London: College Publications, 2015. P. 61–76.
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Czermak, 1977 – Czermak J. A remark on Gentzen’s calculus of sequents // Notre Dame Journal of Formal Logic. 1977. Vol. 18. P. 471–474.
Da Costa, 1963 – Da Costa N.C.A. Sistemas Formais Inconsistentes. Curitiba: Universidade Federal do Paran´a, 1963.
Da Costa, 1974 – Da Costa N.C.A. On the theory of inconsistent formal systems // Notre Dame Journal of Formal Logic. 1974. Vol. 15. No. 4. P. 497–510.
D’Ottaviano, Gomes, 2020 – D’Ottaviano I.M.L., Gomes E.L. Baptizing Paraconsistent Logics: The Unique Touch of Mir´o Quesada // South American Journal of Logic. 2020. Vol. 6. No. 2. P. 249–269.
Da Ré et al., 2020 – Da Ré B., Pailos F., Szmuc D., Teijeiro P. Metainferential duality // Journal of Applied Non-Classical Logics. 2020. Vol. 30. No. 4. P. 312–334.
Da Ré et al., 2024 – Da Ré B., Szmuc D., Corbal´an M.I. Non-reflexive nonsense: proof theory of paracomplete weak Kleene logic // Studia Logica. 2024. Vol. 112. No. 6. P. 1243–1259.
Demey, Smessaert, 2016 – Demey L., Smessaert H. Duality in Logic and Language // Internet Encyclopedia of Philosophy. 2016. URL: https://www.iep.utm.edu/ dual-log/ (дата обращения: 01.07.2024).
Goodman, 1981 – Goodman N.D. The Logic of Contradiction // Mathematical Logic Quarterly. 1981. Vol. 27. No. 8–10. P. 119–126.
Halldén, 1949 – Halldén S. The Logic of Nonsense. Uppsala: Uppsala Universitet, 1949. James, 1996 – James W. Closed Set Logic in Categories. PhD thesis, Department of Philosophy, The University of Adelaide, 1996. URL: https://digital.library. adelaide.edu.au/dspace/bitstream/2440/18746/2/02whole.pdf (дата обращения: 28.10.2024).
Jaśkowski, 1969 – Jaśkowski S. A propositional calculus for inconsistent deductive systems // Studia Logica. 1969. Vol. 24. P. 143–157.
Kapsner, 2014 – Kapsner A. Logics and Falsifications. A New Perspective on Constructivist Semantics. Trends in Logic. Vol. 40. Cham: Springer, 2014.
Kleene, 1952 – Kleene S.C. Introduction to metamathematics. Groningen: Wolters–Noordhoff Publishing, 1952. 560 p.
Loparić, da Costa, 1984 – Loparić A., da Costa N.C.A. Paraconsistency, paracompleteness, and valuations // Logique et Analyse. 1984. Vol. 106. P. 119–131.
Macias et al., 2023 – Macias V.B., Coniglio M.E., Hernández-Tello A. Genuine Paracomplete Logics // Logic Journal of the IGPL. 2023. Vol. 31. No. 5. P. 961–987.
Marcos, 2005a – Marcos J. Nearly every normal modal logic is paranormal // Logique et Analyse. 2005. Vol. 48. No. 189/192. P. 279–300.
Marcos, 2005b – Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic. Vol. 2. Monza: Polimetrica International Scientific Publisher, 2005. P. 53–69.
Middelburg, 2021 – Middelburg C.A. On the strongest three-valued paraconsistent logic contained in classical logic and its dual // Journal of Logic and Computation. 2021. Vol. 31. Iss. 2. P. 597–611.
Miller, 2000 – Miller D. Paraconsistent logic for falsificationists // Proc. 1st Workshop on Logic and Language (Universidad de Sevilla). Sevilla: Editorial Kronos s.a., 2000. P. 197–204.
Mortensen, 1995 – Mortensen C. Inconsistent Mathematics. Dordrecht; Boston; London: Kluwer Academic Publishers, 1995.
Priest, 2002 – Priest G. Paraconsistent Logic // Gabbay D.M., Guenthner F. (eds.) Handbook of Philosophical Logic. Dordrecht: Springer, 2002. P. 287–393.
Priest, 2007 – Priest G. Paraconsistency and Dialetheism // Handbook of the History of Logic. Vol. 8 / D. Gabbay and J. Woods (eds.). Amsterdam, North Holland, 2007. P. 129–204.
Priest, 2009 – Priest G. Dualising Intuitionictic Negation // Principia: An International Journal of Epistemology. 2009. Vol. 13. No. 2. P. 165–184.
Priest et al., 2024 – Priest G., Berto F., Weber Z. Dialetheism // The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Ed. N. Zalta & U. Nodelman (eds.). URL: https://plato.stanford.edu/entries/dialetheism/ (дата обращения: 20.12.2024).
Troelstra, 1990 – Troelstra A.S. On the Early History of Intuitionistic Logic // Mathematical Logic. Petkov P.P. (ed.). Boston, MA: Springer, 1990.
Schröder, 1877 – Schröder E. Der Operationskreis des Logikkalküls. Leipzig: B.G. Teubner, 1877.
Sette, Carnielli, 1995 – Sette A.M., Carnielli W.A. Maximal Weakly-Intuitionistic Logics // Studia Logica. 1995. Vol. 55. P. 181–203.
Shoesmith, Smiley, 1978 – Shoesmith D.J., Smiley T.J. Multiple-Conclusion Logic. Cambridge: Cambridge University Press, 1978. 409 p.
Shramko, 2005 – Shramko Ya. Dual intuitionistic logic and a variety of negations: The logic of scientific research // Studia Logica. 2005. Vol. 80. No. 2–3. P. 347–367.
Stopa, 2022 – Stopa M. Intuitionistic logic versus paraconsistent logic. Categorical approach. A dissertation submitted for the degree of Doctor of Philosophy. Krak´ow: Jagiellonian University, 2022. URl: https://philarchive.org/archive/STOILV (дата обращения: 20.12.2024).
Sylvan, 1990 – Sylvan R. Variations on da Costa C Systems and dual-intuitionistic logics I. Analyses of C ω and CC ω // Studia Logica. 1990. Vol. 49. P. 47–65.
Urbas, 1996 – Urbas I. Dual-intuitionistic logic // Notre Dame Journal of Formal Logic. 1996. Vol. 37. No. 3. P. 440–451.
Гильберт, Аккерман, 1947 – Гильберт Д., Аккерман В. Основы теоретической логики. М., 1947.
Девяткин, 2018 – Девяткин Л.Ю. Многозначные расширения классической логики высказываний. М.: ИФ РАН, 2018.
Карпенко, 2010 – Карпенко А.С. Развитие многозначной логики. М.: Изд-во ЛКИ, 2010.
Клини, 1957 – Клини С.К. Введение в метаматематику. М.: ИЛ, 1957.
Клини, 1973 – Клини С.К. Математическая логика. М.: Мир, 1973.
Марченков, 2014 – Марченков С.С. Основы теории булевых функций. М.: ФИЗМАТЛИТ, 2014.
Томова, 2012 – Томова Н.Е. Естественные трехзначные логики: функциональные свойства и отношения. М.: ИФ РАН, 2012.
Томова, 2022 – Томова Н.Е. К вопросу о критерии паранепротиворечивости логик // Логические исследедования / Logical Investigations. 2022. T. 28. № 2. С. 77–95.
Томова, 2023 – Томова Н.Е. К вопросу о критерии параполноты логик // Логические исследедования / Logical Investigations. 2023. T. 29. № 2. С. 104–124. Чёрч, 1960 – Чёрч А. Введение в математическую логику. Т. 1. М., 1960.
Atiyah, 2007 – Atiyah M. Duality in mathematics and physics. Lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB). P. 69–91. URL: https://fme.upc.edu/ca/arxius/butlleti-digital/riemann/071218_ conferencia_atiyah-d_article.pdf (дата обращения: 01.07.2024).
Binder et al., 2007a – Binder D., Piecha Th., Schroeder-Heister P. (eds.) The Logical Writings of Karl Popper. Trends in Logic. Vol. 58. Cham: Springer, 2022.
Binder et al., 2007b – Binder D., Piecha Th., Schroeder-Heister P. Popper’s Theory of Deductive Logic // The Logical Writings of Karl Popper. Trends in Logic. Vol. 58. Cham: Springer, 2022. P. 1–79.
Brunner, Carnielli, 2005 – Brunner A.B.M., Carnielli W.A. Anti-intuitionism and paraconsistency // Journal of Applied Logic. 2005. Vol. 3. P. 161–184.
Carnielli et al., 2007 – Carnielli W., Coniglio M.E., Marcos J. Logics of Formal Inconsistency // Logics of Formal Inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic. Vol. 14. Dordrecht: Springer, 2007.
Carnielli et al., 2020 – Carnielli W.A., Coniglio M.E., Rodrigues A. Recovery operators, paraconsistency and duality // Logic Journal of the IGPL. 2020. Vol. 28. No. 5. P. 624–656.
Carnielli, Rodrigues, 2016 – Carnielli W., Rodrigues A. Paraconsistency and duality: between ontological and epistemological views // The Logica Yearbook 2015. College Publications, 2016. URL: https://everest.fapemig.br/files/arq_ definitivos/364/APQ-01659-16/APQ-01659-16-Out1.pdf (дата обращения: 01.07.2024).
Carnielli, Rodrigues, 2021 – Carnielli W., Rodrigues A. On epistemic and ontological interpretations of intuitionistic and paraconsistent paradigms // Logic Journal of the IGPL. 2021. Vol. 29. Iss. 4. P. 569–584.
Ciuni, 2015 – Ciuni R. Conjunction in paraconsistent weak Kleene logic // Logica Yearbook 2014 / Ed. by P. Arazim and M. Danc´ak. London: College Publications, 2015. P. 61–76.
Cobreros, 2013 – Cobreros P. Vagueness: Subvaluationism // Philosophy Compass. 2013. Vol. 8/5. P. 472–485.
Czermak, 1977 – Czermak J. A remark on Gentzen’s calculus of sequents // Notre Dame Journal of Formal Logic. 1977. Vol. 18. P. 471–474.
Da Costa, 1963 – Da Costa N.C.A. Sistemas Formais Inconsistentes. Curitiba: Universidade Federal do Paran´a, 1963.
Da Costa, 1974 – Da Costa N.C.A. On the theory of inconsistent formal systems // Notre Dame Journal of Formal Logic. 1974. Vol. 15. No. 4. P. 497–510.
D’Ottaviano, Gomes, 2020 – D’Ottaviano I.M.L., Gomes E.L. Baptizing Paraconsistent Logics: The Unique Touch of Mir´o Quesada // South American Journal of Logic. 2020. Vol. 6. No. 2. P. 249–269.
Da Ré et al., 2020 – Da Ré B., Pailos F., Szmuc D., Teijeiro P. Metainferential duality // Journal of Applied Non-Classical Logics. 2020. Vol. 30. No. 4. P. 312–334.
Da Ré et al., 2024 – Da Ré B., Szmuc D., Corbal´an M.I. Non-reflexive nonsense: proof theory of paracomplete weak Kleene logic // Studia Logica. 2024. Vol. 112. No. 6. P. 1243–1259.
Demey, Smessaert, 2016 – Demey L., Smessaert H. Duality in Logic and Language // Internet Encyclopedia of Philosophy. 2016. URL: https://www.iep.utm.edu/ dual-log/ (дата обращения: 01.07.2024).
Goodman, 1981 – Goodman N.D. The Logic of Contradiction // Mathematical Logic Quarterly. 1981. Vol. 27. No. 8–10. P. 119–126.
Halldén, 1949 – Halldén S. The Logic of Nonsense. Uppsala: Uppsala Universitet, 1949. James, 1996 – James W. Closed Set Logic in Categories. PhD thesis, Department of Philosophy, The University of Adelaide, 1996. URL: https://digital.library. adelaide.edu.au/dspace/bitstream/2440/18746/2/02whole.pdf (дата обращения: 28.10.2024).
Jaśkowski, 1969 – Jaśkowski S. A propositional calculus for inconsistent deductive systems // Studia Logica. 1969. Vol. 24. P. 143–157.
Kapsner, 2014 – Kapsner A. Logics and Falsifications. A New Perspective on Constructivist Semantics. Trends in Logic. Vol. 40. Cham: Springer, 2014.
Kleene, 1952 – Kleene S.C. Introduction to metamathematics. Groningen: Wolters–Noordhoff Publishing, 1952. 560 p.
Loparić, da Costa, 1984 – Loparić A., da Costa N.C.A. Paraconsistency, paracompleteness, and valuations // Logique et Analyse. 1984. Vol. 106. P. 119–131.
Macias et al., 2023 – Macias V.B., Coniglio M.E., Hernández-Tello A. Genuine Paracomplete Logics // Logic Journal of the IGPL. 2023. Vol. 31. No. 5. P. 961–987.
Marcos, 2005a – Marcos J. Nearly every normal modal logic is paranormal // Logique et Analyse. 2005. Vol. 48. No. 189/192. P. 279–300.
Marcos, 2005b – Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic. Vol. 2. Monza: Polimetrica International Scientific Publisher, 2005. P. 53–69.
Middelburg, 2021 – Middelburg C.A. On the strongest three-valued paraconsistent logic contained in classical logic and its dual // Journal of Logic and Computation. 2021. Vol. 31. Iss. 2. P. 597–611.
Miller, 2000 – Miller D. Paraconsistent logic for falsificationists // Proc. 1st Workshop on Logic and Language (Universidad de Sevilla). Sevilla: Editorial Kronos s.a., 2000. P. 197–204.
Mortensen, 1995 – Mortensen C. Inconsistent Mathematics. Dordrecht; Boston; London: Kluwer Academic Publishers, 1995.
Priest, 2002 – Priest G. Paraconsistent Logic // Gabbay D.M., Guenthner F. (eds.) Handbook of Philosophical Logic. Dordrecht: Springer, 2002. P. 287–393.
Priest, 2007 – Priest G. Paraconsistency and Dialetheism // Handbook of the History of Logic. Vol. 8 / D. Gabbay and J. Woods (eds.). Amsterdam, North Holland, 2007. P. 129–204.
Priest, 2009 – Priest G. Dualising Intuitionictic Negation // Principia: An International Journal of Epistemology. 2009. Vol. 13. No. 2. P. 165–184.
Priest et al., 2024 – Priest G., Berto F., Weber Z. Dialetheism // The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Ed. N. Zalta & U. Nodelman (eds.). URL: https://plato.stanford.edu/entries/dialetheism/ (дата обращения: 20.12.2024).
Troelstra, 1990 – Troelstra A.S. On the Early History of Intuitionistic Logic // Mathematical Logic. Petkov P.P. (ed.). Boston, MA: Springer, 1990.
Schröder, 1877 – Schröder E. Der Operationskreis des Logikkalküls. Leipzig: B.G. Teubner, 1877.
Sette, Carnielli, 1995 – Sette A.M., Carnielli W.A. Maximal Weakly-Intuitionistic Logics // Studia Logica. 1995. Vol. 55. P. 181–203.
Shoesmith, Smiley, 1978 – Shoesmith D.J., Smiley T.J. Multiple-Conclusion Logic. Cambridge: Cambridge University Press, 1978. 409 p.
Shramko, 2005 – Shramko Ya. Dual intuitionistic logic and a variety of negations: The logic of scientific research // Studia Logica. 2005. Vol. 80. No. 2–3. P. 347–367.
Stopa, 2022 – Stopa M. Intuitionistic logic versus paraconsistent logic. Categorical approach. A dissertation submitted for the degree of Doctor of Philosophy. Krak´ow: Jagiellonian University, 2022. URl: https://philarchive.org/archive/STOILV (дата обращения: 20.12.2024).
Sylvan, 1990 – Sylvan R. Variations on da Costa C Systems and dual-intuitionistic logics I. Analyses of C ω and CC ω // Studia Logica. 1990. Vol. 49. P. 47–65.
Urbas, 1996 – Urbas I. Dual-intuitionistic logic // Notre Dame Journal of Formal Logic. 1996. Vol. 37. No. 3. P. 440–451.