Journal of Philosophical Logic

, Volume 45, Issue 1, pp 73–87 | Cite as

An Interpretation of Łukasiewicz’s 4-Valued Modal Logic

  • José M. MéndezEmail author
  • Gemma Robles
  • Francisco Salto


A simple, bivalent semantics is defined for Łukasiewicz’s 4-valued modal logic Łm4. It is shown that according to this semantics, the essential presupposition underlying Łm4 is the following: A is a theorem iff A is true conforming to both the reductionist (rt) and possibilist (pt) theses defined as follows: rt: the value (in a bivalent sense) of modal formulas is equivalent to the value of their respective argument (that is, ‘ A is necessary’ is true (false) iff A is true (false), etc.); pt: everything is possible. This presupposition highlights and explains all oddities arising in Łm4.


Many-valued logics Modal logics 4-valued logics Łukasiewicz’s 4-valued modal logic Bivalent semantics 



Work supported by research project FFI2011-28494, financed by the Spanish Ministry of Economy and Competitiveness. -G. Robles is supported by Program Ramón y Cajal of the Spanish Ministry of Economy and Competitiveness. -We sincerely thank the referees of the JPL for their comments and suggestions on a previous draft of this paper.


  1. 1.
    Anderson, A.R., & Belnap, N.D. Jr. (1975). Entailment. The Logic of Relevance and Necessity, vol. I: Princeton University Press.Google Scholar
  2. 2.
    Carnielli, W. (Manuscript). Many-valued models (available at http://www.unilog-org/many-valued.pdf, pp. 1–36.
  3. 3.
    Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University Press.Google Scholar
  4. 4.
    Dugundji, J. (1940). Note on a property of matrices for Lewis and Langford’s calculi of propositions. Journal of Symbolic Logic, 5(4), 150–151.CrossRefGoogle Scholar
  5. 5.
    Font, J.M., & Hajek, P. (2002). On Łukasiewicz four-valued modal logic. Studia Logica, 70(2), 157–182.CrossRefGoogle Scholar
  6. 6.
    González, C. (2012). MaTest, available at (Last access 08/02/2015).
  7. 7.
    Hughes, G.E., & Creswell, M.J. (1968). Introduction to modal logic. London: Methuen.Google Scholar
  8. 8.
    Lemmon, E.J. (1966). Algebraic semantics for modal logics I. The Journal of Symbolic Logic, 31(1), 46–65.CrossRefGoogle Scholar
  9. 9.
    Lemmon, E.J. (1966). Algebraic semantics for modal logics II. The Journal of Symbolic Logic, 31(2), 191–218.CrossRefGoogle Scholar
  10. 10.
    Lewis, C.I., & Langdord, C.H. (1932). Symbolic Logic, New York: Century Company. Reprinted, New York: Dover Publications, 2nd edition, 1959, with a new Appendix III (Final Note on System S2) by Lewis.Google Scholar
  11. 11.
    Łukasiewicz, J. (1951). Aristotle’s syllogistic from the standpoint of modern formal logic. Oxford: Clarendon Press.Google Scholar
  12. 12.
    Łukasiewicz, J (1953). A system of modal logic. The Journal of Computing Systems, 1, 111–149.Google Scholar
  13. 13.
    Mendelson, E. (1964). Introduction to mathematical logic, 5th edn: Chapman and Hall/CRC.Google Scholar
  14. 14.
    Méndez, J.M., & Robles, G. (In preparation). A strong and rich 4-valued modal logic without Łukasiewicz-type paradoxes.Google Scholar
  15. 15.
    Mortensen, C. (1989). Anything is possible. Erkenntnis, 30, 319–337.CrossRefGoogle Scholar
  16. 16.
    Mortensen, C. (2005). It isn’t so, but could it be? Logique et Analyse, 48(189–192), 351–360.Google Scholar
  17. 17.
    Smiley, T.J. (1961). On Łukasiewicz’s Ł-modal system. Notre Dame Journal of Formal Logic, 2, 149–153.CrossRefGoogle Scholar
  18. 18.
    Tkaczyk, M. (2011). On axiomatization of Łukasiewicz’s four-valued modal logic. Logic and Logical Philosophy, 20(3), 215–232.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • José M. Méndez
    • 1
    Email author
  • Gemma Robles
    • 2
  • Francisco Salto
    • 2
  1. 1.Campus Unamuno, Edificio FESUniversidad de SalamancaSalamancaSpain
  2. 2.Departamento de Psicología, Sociología y FilosofíaUniversidad de León Campus de Vegazana, s/nLeónSpain

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