On genuine paraconsistent and genuine paracomplete newline many-valued logics
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Published:
22.12.2019
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Abstract
The paper is devoted to the analysis of several classes of paraconsistent and paracomplete logics using the example of three- and four-valued logics which preserve the classical truth-values. The results presented in the paper can be divided into three groups. The first group concerns paraconsistent logics. We demonstrate that all three-valued genuine paraconsistent logics are logics of formal inconsistency, while all three-valued logics of formal inconsistency which expand the positive fragment of classical propositional logic are linguistic variants of genuine paraconsistent logics.
Further, we provide examples of four-valued genuine paraconsistent logics which are not logics of formal inconsistency.
We present a number of necessary conditions that four-valued matrices of genuine paraconsistent logics must satisfy in order to not induce logics that are logics of formal inconsistency.
The second group concerns paracomplete logics.
We demonstrate that all three-valued genuine paracomplete logics are logics of formal undeterminedness, while all three-valued logics of formal undeterminedness which expand the dual-positive fragment of classical propositional logic are linguistic variants of genuine paracomplete logics.
We also provide examples of four-valued genuine paracomplete logics which are not logics of formal undeterminedness.
We present a number of necessary conditions that four-valued matrices of genuine paracomplete logics must satisfy in order to not induce logics that are logics of formal undeterminedness.
The second group concerns paranormal logics.
We give a number of necessary conditions which must be satisfied by four-valued matrices of genuine paranormal logics so that those logics were neither logics of formal inconsistency nor logics of formal undeterminedness.
For paranormal expansions of Belnap's ``useful four-valued logic'' we also provide the necesssary and sufficient conditions for being maximal genuine paranormal logics which are neither logics of formal inconsistency nor logics of formal undeterminedness.
Further, we provide examples of four-valued genuine paraconsistent logics which are not logics of formal inconsistency.
We present a number of necessary conditions that four-valued matrices of genuine paraconsistent logics must satisfy in order to not induce logics that are logics of formal inconsistency.
The second group concerns paracomplete logics.
We demonstrate that all three-valued genuine paracomplete logics are logics of formal undeterminedness, while all three-valued logics of formal undeterminedness which expand the dual-positive fragment of classical propositional logic are linguistic variants of genuine paracomplete logics.
We also provide examples of four-valued genuine paracomplete logics which are not logics of formal undeterminedness.
We present a number of necessary conditions that four-valued matrices of genuine paracomplete logics must satisfy in order to not induce logics that are logics of formal undeterminedness.
The second group concerns paranormal logics.
We give a number of necessary conditions which must be satisfied by four-valued matrices of genuine paranormal logics so that those logics were neither logics of formal inconsistency nor logics of formal undeterminedness.
For paranormal expansions of Belnap's ``useful four-valued logic'' we also provide the necesssary and sufficient conditions for being maximal genuine paranormal logics which are neither logics of formal inconsistency nor logics of formal undeterminedness.
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How to Cite
Devyatkin L. Y. On genuine paraconsistent and genuine paracomplete newline many-valued logics // Logicheskie Issledovaniya / Logical Investigations. 2019. VOL. 25. № 2. C. 26-45.
Issue
Section
Non-classical logics
Copyright (c) 2019 Леонид Юрьевич Девяткин
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
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Loparic A., da Costa N.C.A. Paraconsistency, paracompleteness, and valuations // Logique et Analyse. Nouvelle serie. 1984. Vol. 27. No. 106. P. 119–131.
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Omori H., Wansing H. 40 years of FDE: an introductory overview // Studia Logica. 2017. Vol. 105. No. 6. P. 1021–1049.
Sette A.M., Carnielli W.A. Maximal weakly-intuitionistic logics // Studia Logica. 1995. Vol. 55. No. 1. P. 181–203.
Urbas I. Paraconsistency // Studies in Soviet Thought. 1990. Vol. 39. No. 3–4. P. 343–354.
Arieli O., Avron A Four-valued paradefinite logics // Studia Logica. 2017. Vol. 105. No. 6. P. 1087–1122.
Arieli O., Avron A, Zamansky A. Maximal and premaximal paraconsistency in the framework of three-valued semantics // Studia Logica. 2011. Vol. 97. No 1. P. 31–60.
Arieli O., Avron A, Zamansky A. Ideal paraconsistent logics. Studia Logica. 2011. Vol. 99. No. 1–3. P. 31–60.
Belnap N. A useful four-valued logic // Modern Uses of Multiple-Valued Logic / Ed. by J.M. Dunn, G. Epstein. Reidel Publishing Co., 1977. P. 8–37.
Beziau J.-Y. The future of paraconsistent logic // Logical Investigations. 1999. Vol. 2. P.1–23.
Beziau J.-Y. What is paraconsistent logic // Frontiers of paraconsistent logic / Ed. by D. Batens et al. Research Studies Press, 2000. P. 95–111.
Beziau J.-Y. Two genuine 3-valued paraconsistent logics // Towards Paraconsistent Engineering / Ed. by S. Akama. Springer, 2016. P. 35–47.
Beziau J.-Y., Franceschetto A. Strong three-valued paraconsistent logics / New Directions in Paraconsistent Logic // Ed. by J.-Y. Beziau et al. Springer, New Delhi, 2015. P. 131–145.
Brunner A.B., Carnielli W.A. Anti-intuitionism and paraconsistency // Journal of Applied Logic. 2005. Vol. 3. No. 1. P. 161–184.
Carnielli W., Coniglio M.E. Paraconsistent Logic: Consistency, Contradiction and Negation. Springer, 2016. 398 p.
Carnielli W., Marcos J. A taxonomy of C-systems // Paraconsistency: The Logical Way to the Inconsistent / Ed. by W.A. Carnielli et al. Marcel Dekker, 2002. P. 1–94.
Carnielli W., Coniglio M.E., Marcos J. Logics of formal inconsistency // Handbook of Philosophical Logic. Vol. 14 / Ed. by D. Gabbay, F. Guenthner. Springer, 2007. P. 1–93.
Karpenko A.S. Four-valued logics BD and DM4: Expansions // Bulletin of the Section of Logic. 2017. Vol. 46. No. 1–2. P. 33–45.
Loparic A., da Costa N.C.A. Paraconsistency, paracompleteness, and valuations // Logique et Analyse. Nouvelle serie. 1984. Vol. 27. No. 106. P. 119–131.
Marcos J. Nearly every normal modal logic is paranormal // Logique et Analyse. Nouvelle serie. 2005. Vol. 48. No. 189–192. P. 279–300.
Omori H., Wansing H. 40 years of FDE: an introductory overview // Studia Logica. 2017. Vol. 105. No. 6. P. 1021–1049.
Sette A.M., Carnielli W.A. Maximal weakly-intuitionistic logics // Studia Logica. 1995. Vol. 55. No. 1. P. 181–203.
Urbas I. Paraconsistency // Studies in Soviet Thought. 1990. Vol. 39. No. 3–4. P. 343–354.