On the question of the criteria for the paracompleteness of logics
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07.11.2023
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Abstract
The paper discusses some aspects related to the definition of a paracomplete logics. There are various ways to formalize the basic condition of paracompleteness – the requirement of the presence in the logical system of such formulas that these formulas themselves and their negations are false. The corresponding definitions of the paracompleteness are given. It is indicated in which cases these definitions may be equivalent. The conditions of equivalence of the law of excluded middle $\varphi\vee\neg\varphi$, the law of Clavius $(\neg\varphi\to\varphi) \to\varphi$ and its special case $(\varphi\to\neg\varphi) \to \neg\varphi$ are considered. The paper deals with issues related to the issues of the paracomplete negation, as well as its modal interpretation.
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How to Cite
Tomova N. On the question of the criteria for the paracompleteness of logics // Logicheskie Issledovaniya / Logical Investigations. 2023. VOL. 29. № 2. C. 104-124.
Issue
Section
Non-classical logics
Copyright (c) 2023 Наталья Евгеньевна Томова
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
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Lenzen, 1998 – Lenzen W. Necessary Conditions for Negation-Operators (with Particular Applications to Paraconsistent Negation) // Besnard P., Hunter A.(eds.). Reasoning with Actual and Potential Contradictions. Handbook of Defeasible Reasoning and Uncertainty Management Systems. Vol. 2. Dordrecht: Springer, 1998. P. 211–239.
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. Lopari´c, da Costa, 1984 – Lopari´c A., da Costa N.C.A. Paracomplete logics which are dual to the paraconsistent logics L3A and L3B // Logique et Analyse. 1984. Vol. 106. P. 119–131.
Lukasiewicz, 1953 – Lukasiewicz J. A system of modal logic // J. Comput. Systems. 1953. Vol. 1. P. 111–149.
Marcos, 2005a – Marcos J. Nearly every normal modal logic is paranormal // Logique et Analyse. Vol. 48. № 189/192. P. 279–300.
Marcos, 2005b – Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic / Ed. by G. Sica. Monza: Polimetrica, 2005. P. 39–55. 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. Is. 2. P. 597–611.
Petrukhin, 2018 – Petrukhin Y. Generalized Correspondence Analysis for Three-Valued Logics // Logica Universalis. 2018. Vol. 12. № 3–4. P. 423–460.
Priest, 1979 – Priest G. The logic of paradox // Journal of Philosophical Logic. 1979. Vol. 8. P. 219–241.
Rosser, Turquette, 1952 – Rosser J.B., Turquette A.R. Many-Valued logics. Amsterdam: North-Holland, 1952. 124 p.
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 University Press, 1978. 409 p.
Smolenov, 1998 – Smolenov H. Paraconsistency, Paracompleteness and Intentional Contradictions // The Journal of Non-Classical Logic. 1997. Vol. 4. № 1. P. 6–35.
Девяткин, 2019 – Девяткин Л.Ю. О подлинно паранепротиворечивых и подлинно параполных многозначных логиках // Логические исследования. 2019. Т. 25. № 2. С. 26–45.
Девяткин и др., 2007 – Девяткин Л.Ю., Карпенко А.С., Попов В.М. Трехзначные характеристические матрицы классической пропозициональной логики // Труды научно-исследовательского семинара Логического центра Института философии РАН. Вып. XVIII. М.: ИФ РАН, 2007. С. 50–62.
Д’Оттавиано, Гомес, 2018 – Д’Оттавиано И.М.Л., Гомес Э.Л. На заре reductio ad absurdum в ранней античной логике // Современная логика: основания, предмет и перспективы развития. М., 2018. С. 187–229.
Томова, 2022 – Томова Н.Е. К вопросу о критерии паранепротиворечивости логик // Логические исследедования / Logical Investigations. 2022. T. 28. № 2. С. 77–95.
Batens et al., 1999 – Batens D., De Clercq K., Kurtonina N. Embedding and interpolation for some paralogics. The propositional case // Rep. Math. Log. 1999. Vol. 33. P. 29–44. B´eziau, 2000 – B´eziau J.-Y. What is paraconsistent logic? // Frontiers of Paraconsistent Logic. Batens D. et al. (eds.). Research Studies Press, Baldock, 2000. P. 95–111.
Beziau, 2002 – Beziau J.-Y. Are paraconsistent negations negations? // Paraconsistency: The Logical Way to the Inconsistent. W.A. Carnielli et al. (eds.). New York, 2002.
Beziau, 2003 – Beziau J.-Y. New light on the square of oppositions and its nameless corner // Логические исследования. 2003. Т. 10. C. 218–232.
Beziau, 2005 – Beziau J.-Y. Paraconsistent logic from a modal viewpoint // Journal of Applied Logic. 2005. Vol. 3. P. 7–14.
B´eziau, 2016 – B´eziau J.-Y. Two Genuine 3-Valued Paraconsistent Logics // Akama S. (ed.). Towards Paraconsistent Engineering. Intelligent Systems Reference Library. Vol. 110. Cham: Springer, 2016. P. 35–47.
Caret, 2017 – Caret C. Hybridized Paracomplete and Paraconsistent Logics // The Australasian Journal of Logic. 2017. Vol. 14. № 1. P. 281–325.
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. № 5. P. 624–656.
Ciuciura, 2015 – Ciuciura J. A Weakly-Intuitionistic Logic I1 // Logical Investigations. 2015. Vol. 21. No. 2. P. 53–60.
Ciuciura, 2019 – Ciuciura J. Paraconsistency and Paracompleteness // Logical Investigations. 2019. Vol. 25. № 2. P. 46–60.
D’Ottaviano, da Costa, 1970 – D’Ottaviano I.M.L., da Costa N.C.A. Sur un probl`eme de Ja´skowski // Comptes Rendus de l’Acad´emie de Sciences de Paris. Ser. A. 1970. Vol. 270. P. 1349–1353.
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. № 2. P. 249–269.
Hern´andez-Tello et al., 2020 – Hern´andez-Tello A., Borja-Macias V., Coniglio M.E. Paracomplete Logics Dual to the Genuine Paraconsistent Logics: The Three-valued Case // Electronic Notes in Theoretical Computer Science. 2020. Vol. 354. P. 61–74.
Heyting, 1930 – Heyting A. Die formalen Regeln der intuitionistischen Logik // Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin. 1930. P. 42–56. (Англ. пер. в: Mancosu P. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, 1998).
Hyde, 2008 – Hyde D. Vagueness, logic and ontology. Hampshire: Ashgate, 2008.
Lenzen, 1998 – Lenzen W. Necessary Conditions for Negation-Operators (with Particular Applications to Paraconsistent Negation) // Besnard P., Hunter A.(eds.). Reasoning with Actual and Potential Contradictions. Handbook of Defeasible Reasoning and Uncertainty Management Systems. Vol. 2. Dordrecht: Springer, 1998. P. 211–239.
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. Lopari´c, da Costa, 1984 – Lopari´c A., da Costa N.C.A. Paracomplete logics which are dual to the paraconsistent logics L3A and L3B // Logique et Analyse. 1984. Vol. 106. P. 119–131.
Lukasiewicz, 1953 – Lukasiewicz J. A system of modal logic // J. Comput. Systems. 1953. Vol. 1. P. 111–149.
Marcos, 2005a – Marcos J. Nearly every normal modal logic is paranormal // Logique et Analyse. Vol. 48. № 189/192. P. 279–300.
Marcos, 2005b – Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic / Ed. by G. Sica. Monza: Polimetrica, 2005. P. 39–55. 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. Is. 2. P. 597–611.
Petrukhin, 2018 – Petrukhin Y. Generalized Correspondence Analysis for Three-Valued Logics // Logica Universalis. 2018. Vol. 12. № 3–4. P. 423–460.
Priest, 1979 – Priest G. The logic of paradox // Journal of Philosophical Logic. 1979. Vol. 8. P. 219–241.
Rosser, Turquette, 1952 – Rosser J.B., Turquette A.R. Many-Valued logics. Amsterdam: North-Holland, 1952. 124 p.
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 University Press, 1978. 409 p.
Smolenov, 1998 – Smolenov H. Paraconsistency, Paracompleteness and Intentional Contradictions // The Journal of Non-Classical Logic. 1997. Vol. 4. № 1. P. 6–35.