Studia Logica

, Volume 97, Issue 1, pp 31–60 | Cite as

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

  • Ofer ArieliEmail author
  • Arnon Avron
  • Anna Zamansky


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.


paraconsistent logics three-valued logics non-deterministic semantics 


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  1. 1.
    Anderson, A., and N. Belnap, Entailment, vol. 1, Princeton University Press, 1975.Google Scholar
  2. 2.
    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.Google Scholar
  3. 3.
    Arieli O., Zamansky A.: ‘Distance-based non-deterministic semantics for reasoning with uncertainty’. Logic Journal of the IGPL 17(4), 325–350 (2009)CrossRefGoogle Scholar
  4. 4.
    Avron A.: ‘Relevant entailment - Semantics and formal systems’. Journal of Symbolic Logic 49, 334–342 (1984)CrossRefGoogle Scholar
  5. 5.
    Avron A.: ‘Natural 3-valued logics: Characterization and proof theory’. Journal of Symbolic Logic 56 1, 276–294 (1991)CrossRefGoogle Scholar
  6. 6.
    Avron A.: ‘On the expressive power of three-valued and four-valued languages’. Journal of Logic and Computation 9(6), 977–994 (1999)CrossRefGoogle Scholar
  7. 7.
    Avron A.: ‘Combining classical logic, paraconsistency and relevance’. Journal of Applied Logic 3, 133–160 (2005)CrossRefGoogle Scholar
  8. 8.
    Avron A.: ‘Non-deterministic semantics for logics with a consistency operator’. Journal of Approximate Reasoning 45, 271–287 (2007)CrossRefGoogle Scholar
  9. 9.
    Avron A., Konikowska B.: ‘Multi-valued calculi for logics based on nondeterminism’. Logic Journal of the IGPL 13 4, 365–387 (2005)CrossRefGoogle Scholar
  10. 10.
    Avron A., Lev I.: ‘Non-deterministic multi-valued structures’. Journal of Logic and Computation 15, 241–261 (2005)CrossRefGoogle Scholar
  11. 11.
    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.Google Scholar
  12. 12.
    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.Google Scholar
  13. 13.
    Batens, D., ‘Paraconsistent extensional propositional logics’, Logique et Analyse, 90/91 (1980), 195–234.Google Scholar
  14. 14.
    Batens D., De Clercq K., Kurtonina N.: ‘Embedding and interpolation for some paralogics The propositional case’. Reports on Mathematical Logic 33, 29–44 (1999)Google Scholar
  15. 15.
    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.Google Scholar
  16. 16.
    Carnielli W., Marcos J., de Amo S.: ‘Formal inconsistency and evolutionary databases’. Logic and logical philosophy 8, 115–152 (2000)Google Scholar
  17. 17.
    da Costa N.: ‘On the theory of inconsistent formal systems’. Notre Dame Journal of Formal Logic 15, 497–510 (1974)CrossRefGoogle Scholar
  18. 18.
    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.Google Scholar
  19. 19.
    D’Ottaviano I.: ‘The completeness and compactness of a three-valued first-order logic’. Revista Colombiana de Matematicas XIX(1-2), 31–42 (1985)Google Scholar
  20. 20.
    Gabbay, D., and H. Wansing, (eds.), What is Negation?, vol. 13 of Applied Logic Series, Springer, 1999.Google Scholar
  21. 21.
    Gottwald, S., ‘A treatise on many-valued logics’, in Studies in Logic and Computation, vol. 9, Research Studies Press, Baldock, 2001.Google Scholar
  22. 22.
    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.Google Scholar
  23. 23.
    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.Google Scholar
  24. 24.
    Kleene, S. C., Introduction to Metamathematics, Van Nostrand, 1950.Google Scholar
  25. 25.
    Malinowski, G., Many-Valued Logics, Clarendon Press, 1993.Google Scholar
  26. 26.
    Marcos, J., ‘8K solutions and semi-solutions to a problem of da Costa’, Submitted.Google Scholar
  27. 27.
    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.Google Scholar
  28. 28.
    Marcos J.: ‘On negation: Pure local rules’. Journal of Applied Logic 3(1), 185–219 (2005)CrossRefGoogle Scholar
  29. 29.
    Parks R.: ‘A note on R-mingle and Sobociński three-valued logic’. Notre Dame Journal of Formal Logic 13, 227–228 (1972)CrossRefGoogle Scholar
  30. 30.
    Priest G.: ‘Reasoning about truth’. Artificial Intelligence 39, 231–244 (1989)CrossRefGoogle Scholar
  31. 31.
    Sette A.M.: ‘On propositional calculus P1’. Mathematica Japonica 16, 173–180 (1973)Google Scholar
  32. 32.
    Shoesmith D.J., Smiley T.J.: ‘Deducibility and many-valuedness’. Journal of Symbolic Logic 36, 610–622 (1971)CrossRefGoogle Scholar
  33. 33.
    Shoesmith, D. J., and T. J. Smiley, Multiple Conclusion Logic, Cambridge University Press, 1978.Google Scholar
  34. 34.
    Sobociński B.: ‘Axiomatization of a partial system of three-value calculus of propositions’. Journal of Computing Systems 1, 23–55 (1952)Google Scholar
  35. 35.
    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.Google Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.School of Computer ScienceThe Academic College of Tel-AvivTel-AvivIsrael
  2. 2.School of Computer ScienceTel-Aviv UniversityTel-AvivIsrael
  3. 3.Department of Software EngineeringJerusalem College of EngineeringJerusalemIsrael

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