On the question of the criteria for the paraconsistency of logics
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11.01.2023
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Abstract
The paper discusses various aspects related to the definition of paraconsistent logics. The criteria of the paraconsistency of logical systems, which were proposed by S.Jaskowski and N. da Costa, are given. Various formulations of the principle <<from contradiction, anything follows>> ($\textit{ex falso quodlibet}$) and the corresponding definitions of paraconsistent logic are given. It is indicated in which cases these definitions may be equivalent. The problem of explosiveness of the consequence relation with respect to some operators and bundles is also described, and the solutions that have been proposed by various researchers are given. The paper deals with issues related to the paraconsistent negation, the properties of classical negation that are incompatible with the rejection of the principle of <<from contradiction, anything follows>> are indicated.
Different views on the necessity for non-verifiability the principle of non-contradiction in paraconsistent logics are regarded.
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How to Cite
Tomova N. E. On the question of the criteria for the paraconsistency of logics // Logicheskie Issledovaniya / Logical Investigations. 2023. VOL. 28. № 2. C. 77-95.
Issue
Section
Non-classical logics
Copyright (c) 2022 Наталья Евгеньевна Томова
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
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Ciuciura, 1999 – Ciuciura J. History and development of the discursive logic // Logica Trianguli. 1999. Vol. 3. P. 3–31.
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Johansson, 1936 – Johansson I. Der Minimalkalkil, ein reduzierter intuitionistischer Formalismus // Compositio Mathematica. 1936. Vol. 4. P. 119–136.
Karpenko, 1999 – Karpenko A.S. Ja´skowski’s criterion and three-valued paraconsistent logics // Log. Log. Philos. 1999. Vol. 7. P. 81–86.
Kotas, da Costa, 1978 – Kotas J., da Costa N.C.A. On the problem of Ja´skowski and the logics of Lukasiewicz // Mathematical logic. Proc. of the first Brazilian Conf. New York and Basel: Marcel Dekker, inc., 1978. P. 127–139.
Lenzen, 1996 – Lenzen W. Necessary conditions for negation operators // Negation. A Notion in Focus. Ed. by H. Wansing. Berlin: de Gruyter, 1996. P. 37–58.
Lewin, Mikenberg, 2006 – Lewin R.A., Mikenberg I.F. Literal-paraconsistent and literal-paracomplete matrices // Math. Log. Quart. 2006. Vol. 52. No. 5. P. 478–493.
Marcos, 2005a – Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic. Vol. 2. Polimetrica International Scientific Publisher Monza/Italy, 2005. P. 53–69.
Marcos, 2005b – Marcos J. Logics of Formal Inconsistency. PhD Thesis. Campinas, 2005.
da Costa, 1974 – da Costa N.C.A. On the theory of inconsistent formal systems // Notre Dame Journal of Formal Logic. 1974. Vol. 11. No. 4. P. 497–510.
D’Ottaviano, da Costa, 1970 – D’Ottaviano I.M.L., da Costa N.C.A. Sur un probl´eme de Ja´skowski // Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences. S´eries A–B. 1970. Vol. 270. No. 21. P. 1349–1353.
Priest et al., 2014 – Priest G., Routley R., Norman J. (eds.). Paraconsistent Logic. Essays on the Inconsistent. Munchen; Hamden, Wien: Philosophia, 1989. 715 p.
Priest et al., 2014 – Priest G., Tanaka K., Weber Z. Paraconsistent Logic // The Stanford Encyclopedia of Philosophy (Spring 2022 Edition), E.N. Zalta (ed.). URL: https://plato.stanford.edu/archives/spr2022/entries/logic-paraconsistent (дата обращения: 01.06.2022).
Tarafder, Chakraborty, 2014 – Tarafder S., Chakraborty M.Kr. A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory // New Directions in Paraconsistent Logic, ed. by J.-Y. Beziau, M. Chakraborty, S. Dutta. Springer, India, 2015. P. 165–184.
Urbas, 1988 – Urbas I. Paraconsistency and the J-systems of Arruda and da Costa // Logique et Analyse. 1988. Vol. 31. No. 121/122. P. 27–44.
Urbas, 1990 – Urbas I. Paraconsistency // Studies in Soviet Thoughts. 1990. Vol. 39. No. 3–4. P. 343–354.
Девяткин, 2021 – Девяткин Л.Ю. О выразительных возможностях максимально паранепротиворечивых и максимально параполных четырехзначных расширений FDE // Логические исследедования / Logical Investigations. 2021. T. 27. № 2. С. 66–92.
Ишмуратов и др., 1989 – Ишмуратов А.Т., Карпенко А.С., Попов В.М. О паранепротиворечивой логике // Синтаксические и семантические исследования неэкстенсиональных логик. М.: Наука, 1989. С. 261–284.
Карпенко, Томова, 2016 – Карпенко А.С., Томова Н.Е. Трехзначная логика Бочвара и литеральные паралогики. М.: ИФ РАН, 2016. 110 с.
Кварталова, 2004 – Кварталова Н.Л. Паранепротиворечивость и релевантность:дис. . . . канд. филос. наук: 09.00.07. М., 2004. 86 c.
Розоноэр, 1983 – Розоноэр Л.И. О выявлении противоречий в формальных теориях. I // Автоматтика и телемеханика. 1983. Вып. 6. С. 113–124.
Arieli et al., 2011 – Arieli O., Avron A., Zamansky A. Ideal Paraconsistent Logics // Studia Logica. 2011. Vol. 99. P. 31–60.
Arruda, da Costa, 1974 – Arruda A.I., da Costa N.C.A. Le sch´ema de la s´eparation et les calculs J n // Mathematiea Japonieae. 1974. Vol. 19. P. 183–186.
Basu, Roy, 2022 – Basu S., Roy S. Negation-Free Definitions of Paraconsistency // A. Indrzejczak, M. Zawidzki (eds.). 10th International Conference on Non-Classical Logics. Theory and Applications (NCL 2022) EPTCS 358, 2022. P. 150–159.
Batens, 1980 – Batens D. Paraconsistent extentional propositional logics // Logique et Analyse. 1980. Vol. 90–91. P. 127–139.
Batens, 2000 – Batens D. A survey of inconsistency-adaptive logics. 2000.
Beziau, 1999 – Beziau J.-Y. The future of paraconsistent logic // Logical Studies. 1999. Vol. 2. P. 1–23.
Beziau, 2000 – Beziau J.-Y. What is paraconsistent logic? // Frontiers of Paraconsistent Logic. Batens D. et al. (eds.). Research Studies Press, Baldock, 2000. P. 95–111.
Beziau, 1999 – Beziau J.-Y. Are paraconsistent negations negations? // W.A. Carnielli, M.E. Coniglio, and I.M.L. D’Ottaviano (eds.), Paraconsistency: The Logical Way to the Inconsistent, Proceedings of the II World Congress on Paraconsistency, held in Juquehy, BR, May 8–12, 2000. Vol. 228 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, 2002. P. 465–486.
Beziau, 2016 – Beziau J.-Y. Two Genuine 3-Valued Paraconsistent Logics // Akama S. (ed.). Towards Paraconsistent Engineering. Intelligent Systems Reference Library. Vol. 110. Springer Cham, 2016. P. 35–47.
Ciuciura, 1999 – Ciuciura J. History and development of the discursive logic // Logica Trianguli. 1999. Vol. 3. P. 3–31.
D’Ottaviano, da Costa, 1970 – D’Ottaviano I.M.L., da Costa N.C.A. Sur un probl´eme de Ja´skowski // C.R. Acad. Sc. Paris. 1970. 270A. P. 1349–1353.
Ja´skowski, 1969 – Ja´skowski S. A propositional calculus for inconsistent deductive systems // Studia Logica. 1969. Vol. 24. P. 143–157.
Johansson, 1936 – Johansson I. Der Minimalkalkil, ein reduzierter intuitionistischer Formalismus // Compositio Mathematica. 1936. Vol. 4. P. 119–136.
Karpenko, 1999 – Karpenko A.S. Ja´skowski’s criterion and three-valued paraconsistent logics // Log. Log. Philos. 1999. Vol. 7. P. 81–86.
Kotas, da Costa, 1978 – Kotas J., da Costa N.C.A. On the problem of Ja´skowski and the logics of Lukasiewicz // Mathematical logic. Proc. of the first Brazilian Conf. New York and Basel: Marcel Dekker, inc., 1978. P. 127–139.
Lenzen, 1996 – Lenzen W. Necessary conditions for negation operators // Negation. A Notion in Focus. Ed. by H. Wansing. Berlin: de Gruyter, 1996. P. 37–58.
Lewin, Mikenberg, 2006 – Lewin R.A., Mikenberg I.F. Literal-paraconsistent and literal-paracomplete matrices // Math. Log. Quart. 2006. Vol. 52. No. 5. P. 478–493.
Marcos, 2005a – Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic. Vol. 2. Polimetrica International Scientific Publisher Monza/Italy, 2005. P. 53–69.
Marcos, 2005b – Marcos J. Logics of Formal Inconsistency. PhD Thesis. Campinas, 2005.
da Costa, 1974 – da Costa N.C.A. On the theory of inconsistent formal systems // Notre Dame Journal of Formal Logic. 1974. Vol. 11. No. 4. P. 497–510.
D’Ottaviano, da Costa, 1970 – D’Ottaviano I.M.L., da Costa N.C.A. Sur un probl´eme de Ja´skowski // Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences. S´eries A–B. 1970. Vol. 270. No. 21. P. 1349–1353.
Priest et al., 2014 – Priest G., Routley R., Norman J. (eds.). Paraconsistent Logic. Essays on the Inconsistent. Munchen; Hamden, Wien: Philosophia, 1989. 715 p.
Priest et al., 2014 – Priest G., Tanaka K., Weber Z. Paraconsistent Logic // The Stanford Encyclopedia of Philosophy (Spring 2022 Edition), E.N. Zalta (ed.). URL: https://plato.stanford.edu/archives/spr2022/entries/logic-paraconsistent (дата обращения: 01.06.2022).
Tarafder, Chakraborty, 2014 – Tarafder S., Chakraborty M.Kr. A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory // New Directions in Paraconsistent Logic, ed. by J.-Y. Beziau, M. Chakraborty, S. Dutta. Springer, India, 2015. P. 165–184.
Urbas, 1988 – Urbas I. Paraconsistency and the J-systems of Arruda and da Costa // Logique et Analyse. 1988. Vol. 31. No. 121/122. P. 27–44.
Urbas, 1990 – Urbas I. Paraconsistency // Studies in Soviet Thoughts. 1990. Vol. 39. No. 3–4. P. 343–354.