Advertisement

Studia Logica

, Volume 103, Issue 1, pp 91–112 | Cite as

The Logic of Generalized Truth Values and the Logic of Bilattices

  • Sergei P. OdintsovEmail author
  • Heinrich Wansing
Article

Abstract

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \({\models_t}\) and \({\models_f}\) , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 (Shramko and Wansing, J Philos Logic, 34:121–153, 2005). The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and distributive bilattices with conflation (Rivieccio, 2010).

Keywords

Bilattices Trilattices Generalized truth values Twist-structures Axiomatization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, A. R., N. D. Belnap., and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, Vol. II, Princeton University Press, Princeton, NJ, 1992.Google Scholar
  2. 2.
    Arieli O., Avron A.: Reasoning with logical bilattices. Journal of Logic, Language, and Information 5, 25–63 (1996)CrossRefGoogle Scholar
  3. 3.
    Belnap, N. D., A useful four-valued logic, in G. Epstein, and M. J. Dunn (eds.), Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 1977, pp. 7–37.Google Scholar
  4. 4.
    Belnap, N. D., How computer should think, in G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press, Stocksfield, 1977, pp. 30–56.Google Scholar
  5. 5.
    Bou, F., R. Jansana, and U. Rivieccio, Varieties of interlaced bilattices, Algebra Universalis 66:115–141, 2011.Google Scholar
  6. 6.
    Burris, S., H. P. and Sankappanavar, A Course in Universal Algebra, Milennium Edition, available at http://orion.math.iastate.edu/cliff/BurrisSanka.pdf.
  7. 7.
    Fidel, M. M., An algebraic study of a propositional system of Nelson, in Mathematical Logic, Proceedings of the First Brasilian Conference, Campinas 1977, Lecture Notes Pure Applied Mathematics, Vol. 39, 1978, pp. 99–117.Google Scholar
  8. 8.
    Fitting M.: Bilattices and the semantics of logic programming. Journal of Logic Programming 11(1–2), 91–116 (1991)CrossRefGoogle Scholar
  9. 9.
    Fitting M.: The family of stable models. Journal of Logic Programming 17(2–4), 197–225 (1993)CrossRefGoogle Scholar
  10. 10.
    Fitting M.: Kleene’s three valued logics and their children. Fundamenta Informaticae 20, 113–131 (1994)Google Scholar
  11. 11.
    Fitting M.: A theory of truth that prefers falsehood. Journal of Philosophical Logic 26, 477–500 (1997)CrossRefGoogle Scholar
  12. 12.
    Fitting M.: Fixpoint semantics for logic programming—A survey. Theoretical Computer Science 278(1–2), 25–51 (2002)CrossRefGoogle Scholar
  13. 13.
    Fitting, M., Bilattices are nice things, in T. Bolander, V. Hendricks, and S. A.Pedersen (eds.), Self-reference, CSLI Lecture Notes, CSLI, Stanford, 2006, pp. 53–77.Google Scholar
  14. 14.
    Gargov G.: Knowledge, uncertainty and ignorance in logic: Bilattices and beyond. Journal of Applied Non-classical Logics 9(2–3), 195–283 (1999)CrossRefGoogle Scholar
  15. 15.
    Ginsberg, M., Multi-valued logics, in Proceedings AAAI-86, Fifth National Conference on Artificial Intelligence, Morgan Kaufman, Los Altos, 1986, pp. 243–247.Google Scholar
  16. 16.
    Ginsberg M.: Multivalued logics: A uniform approach to reasoning in AI. Computational Intelligence 4, 256–316 (1988)Google Scholar
  17. 17.
    Kracht M.: On extensions of intermediate logics by strong negation. Journal of Philosophical Logic 27, 49–73 (1998)CrossRefGoogle Scholar
  18. 18.
    Miura S.: A remark on the intersection of two logics. Nagoya Mathematical Journal 26, 167–171 (1966)Google Scholar
  19. 19.
    Odintsov, S. P., Algebraic semantics for paraconsistent Nelson’s Logic, Journal of Logic and Computation 13:453–468, 2003.Google Scholar
  20. 20.
    Odintsov, S. P., Constructive Negations and Paraconsistency, Springer, Dordrecht, 2008.Google Scholar
  21. 21.
    Odintsov S.P.: On axiomatizing Shramko-Wansing’s logic. Studia Logica 91, 407–428 (2009)CrossRefGoogle Scholar
  22. 22.
    Rivieccio, U., An Algebraic Study of Bilattice-based Logics, Ph.D. Dissertation, University of Barcelona, 2010.Google Scholar
  23. 23.
    Rivieccio U.: Representation of interlaced trilattices. Journal of Applied Logic 11, 174–189 (2013)CrossRefGoogle Scholar
  24. 24.
    Shramko, Y., J. M. Dunn, and T. Takenaka, The trilaticce of constructive truth values, Journal of Logic and Computation 11:761–788, 2001.Google Scholar
  25. 25.
    Shramko, Y., and H. Wansing, Some useful 16-valued logics: How a computer network should think, Journal of Philosophical Logic 34:121–153, 2005.Google Scholar
  26. 26.
    Shramko Y., Wansing H.: Hypercontradictions, generalized truth values, and logics of truth and falsehood. Journal of Logic, Language and Information 15, 403–424 (2006)CrossRefGoogle Scholar
  27. 27.
    Shramko, Y., and H. Wansing, Truth and Falsehood. An Inquiry into Generelized Logical Values, Springer, Dordrecht, 2011.Google Scholar
  28. 28.
    Vakarelov D.: Notes on N-lattices and constructive logic with strong negation. Studia logica 36, 109–125 (1977)CrossRefGoogle Scholar
  29. 29.
    Wansing H., Belnap N.: Generalized truth values A reply to Dubois. Logic Journal of the Interest Group in Pure and Applied Logic 18, 921–935 (2009)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Department of Philosophy IIRuhr UniversityBochumGermany

Personalised recommendations