Abstract
Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core” of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.
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References
Anderson, A., and N. Belnap, Entailment, vol. 1, Princeton University Press, 1975.
Arieli O., A. Avron, A. Zamansky, ‘Maximally paraconsistent three-valued logics’, in F. Lin, U. Sattler, and M. Truszczynski, (eds.), Proceedings of the 12th International Conference on Principles of Knowledge Representation and Reasoning (KR’10), AAAI Press, 2010, pp. 310–318.
Arieli O., Zamansky A.: ‘Distance-based non-deterministic semantics for reasoning with uncertainty’. Logic Journal of the IGPL 17(4), 325–350 (2009)
Avron A.: ‘Relevant entailment - Semantics and formal systems’. Journal of Symbolic Logic 49, 334–342 (1984)
Avron A.: ‘Natural 3-valued logics: Characterization and proof theory’. Journal of Symbolic Logic 56 1, 276–294 (1991)
Avron A.: ‘On the expressive power of three-valued and four-valued languages’. Journal of Logic and Computation 9(6), 977–994 (1999)
Avron A.: ‘Combining classical logic, paraconsistency and relevance’. Journal of Applied Logic 3, 133–160 (2005)
Avron A.: ‘Non-deterministic semantics for logics with a consistency operator’. Journal of Approximate Reasoning 45, 271–287 (2007)
Avron A., Konikowska B.: ‘Multi-valued calculi for logics based on nondeterminism’. Logic Journal of the IGPL 13 4, 365–387 (2005)
Avron A., Lev I.: ‘Non-deterministic multi-valued structures’. Journal of Logic and Computation 15, 241–261 (2005)
Avron, A., and A. Zamansky, ‘Many-valued non-deterministic semantics for firstorder logics of formal inconsistency’, in S. Aguzzoli, A. Ciabattoni, B. Gerla, C. Manara, and V.Marra, (eds.), Algebraic and Proof-Theoretic Aspects of Non-classical Logics, LNCS 4460, Springer, 2007, pp. 1–24.
Avron, A., and A. Zamansky, ‘Non-deterministic semantics for logical systems – A survey’, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, Kluwer, 2010. To appear.
Batens, D., ‘Paraconsistent extensional propositional logics’, Logique et Analyse, 90/91 (1980), 195–234.
Batens D., De Clercq K., Kurtonina N.: ‘Embedding and interpolation for some paralogics The propositional case’. Reports on Mathematical Logic 33, 29–44 (1999)
Carnielli, W., M. Coniglio, and J. Marcos, ‘Logics of formal inconsistency’, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 14, Springer, 2007, pp. 1–93. Second edition.
Carnielli W., Marcos J., de Amo S.: ‘Formal inconsistency and evolutionary databases’. Logic and logical philosophy 8, 115–152 (2000)
da Costa N.: ‘On the theory of inconsistent formal systems’. Notre Dame Journal of Formal Logic 15, 497–510 (1974)
Decker, H., ‘A case for paraconsistent logic as foundation of future information systems’, in J. Castro, and E. Teniente, (eds.), Proceedings of the CAiSE Workshops, vol. 2, 2005, pp. 451–461.
D’Ottaviano I.: ‘The completeness and compactness of a three-valued first-order logic’. Revista Colombiana de Matematicas XIX(1-2), 31–42 (1985)
Gabbay, D., and H. Wansing, (eds.), What is Negation?, vol. 13 of Applied Logic Series, Springer, 1999.
Gottwald, S., ‘A treatise on many-valued logics’, in Studies in Logic and Computation, vol. 9, Research Studies Press, Baldock, 2001.
Jaśkowski, S., ‘On the discussive conjunction in the propositional calculus for inconsistent deductive systems’, Logic, Language and Philosophy, 7 (1999), 57–59. Translation of the original paper from 1949.
Karpenko, A., ‘A maximal paraconsistent logic: The combination of two threevalued isomorphs of classical propositional logic.’, in D. Batens, C. Mortensen, G. Priest, and J. Van Bendegem, (eds.), Frontiers of Paraconsistent Logic, vol. 8 of Studies in Logic and Computation, Research Studies Press, 2000, pp. 181–187.
Kleene, S. C., Introduction to Metamathematics, Van Nostrand, 1950.
Malinowski, G., Many-Valued Logics, Clarendon Press, 1993.
Marcos, J., ‘8K solutions and semi-solutions to a problem of da Costa’, Submitted.
Marcos, J., ‘On a problem of da Costa’, in G Sica, (ed.), Essays on the Foundations of Mathematics and Logic, vol. 2, Polimetrica, 2005, pp. 39–55.
Marcos J.: ‘On negation: Pure local rules’. Journal of Applied Logic 3(1), 185–219 (2005)
Parks R.: ‘A note on R-mingle and Sobociński three-valued logic’. Notre Dame Journal of Formal Logic 13, 227–228 (1972)
Priest G.: ‘Reasoning about truth’. Artificial Intelligence 39, 231–244 (1989)
Sette A.M.: ‘On propositional calculus P1’. Mathematica Japonica 16, 173–180 (1973)
Shoesmith D.J., Smiley T.J.: ‘Deducibility and many-valuedness’. Journal of Symbolic Logic 36, 610–622 (1971)
Shoesmith, D. J., and T. J. Smiley, Multiple Conclusion Logic, Cambridge University Press, 1978.
Sobociński B.: ‘Axiomatization of a partial system of three-value calculus of propositions’. Journal of Computing Systems 1, 23–55 (1952)
Urquhart, A., ‘Many-valued logic’, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. II, Kluwer, 2001, pp. 249–295. Second edition.
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Arieli, O., Avron, A. & Zamansky, A. Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics. Stud Logica 97, 31–60 (2011). https://doi.org/10.1007/s11225-010-9296-9
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DOI: https://doi.org/10.1007/s11225-010-9296-9