Advertisement

Journal of Philosophical Logic

, Volume 43, Issue 2–3, pp 303–332 | Cite as

Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

  • Gemma Robles
  • Francisco Salto
  • José M. Méndez
Article
  • 109 Downloads

Abstract

Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent.

Keywords

Many-valued logic Łukasiewicz 3-valued logic Two-valued under-determined and over-determined interpretations Paraconsistent logics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avron, A. (1991). Natural 3-valued logics. characterization and proof theory. Journal of Symbolic Logic, 56, 276–294.CrossRefGoogle Scholar
  2. 2.
    Baaz, M., & Zach, R. (1998). Compact propositional Gӧdel logics. In 28th international symposium on multiple-valued logic proceedings, (pp. 108–113). Fukuoka, Japan.Google Scholar
  3. 3.
    Baaz, M., Preining, N., Zach, R. (2007). First-order Gödel logics. Annals of Pure and Applied Logic, 147, 23–47.CrossRefGoogle Scholar
  4. 4.
    Bimbó, K., & Dunn, J.M. (2008). Generalized Galois logics. Relational semantics of non-classical logic calculi. Stanford, CA: CSLI Publications.Google Scholar
  5. 5.
    Brady, R. (1982). Completeness proofs for the systems RM3 and BN4. Logique et Analyse, 25, 9–32.Google Scholar
  6. 6.
    Brady, R. (Ed.) (2003). Relevant logics and their rivals, Vol. II. Ashgate.Google Scholar
  7. 7.
    Carnielli, W., Coniglio, M., Marcos, J. (2007). Logics of formal inconsistency In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 14, pp. 1–93). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  8. 8.
    Dunn, J.M. (1966). The algebra of intensional logics. Doctoral dissertation, University of Pittsburg (Ann Arbor, University Microfilms).Google Scholar
  9. 9.
    Dunn, J.M. (1976). Intuitive semantics for first-degree entailments and ‘coupled trees’. Philosophical Studies, 29, 149–168.CrossRefGoogle Scholar
  10. 10.
    Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 65, 5–40.CrossRefGoogle Scholar
  11. 11.
    Gabbay, D., & Guenthner, F. (2001). Handbook of philosophical logic, 2nd edn. Dordrecht: Kluwer Academic Publishers.Google Scholar
  12. 12.
    González, C. (2012). MaTest. Available at http://ceguel.es/matest. Last access 26 November 2012.
  13. 13.
    Hähnle, R. (2002). Advanced many-valued logics. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 2, pp. 297–395). Dordrecht: Kluwer Academic Publishers.Google Scholar
  14. 14.
    Karpenko, A.S. (2006). Łukasiewicz logics and prime numbers. Beckington: Luniver Press.Google Scholar
  15. 15.
    Łukasiewicz, J. (1920). On three-valued logic. In J. Łukasiewicz (1970) (pp. 87–88).Google Scholar
  16. 16.
    Łukasiewicz, J. (1970). Selected works. Amsterdam: North-Holland.Google Scholar
  17. 17.
    Łukasiewicz, J., & Tarski, A. (1930). Investigations into the sentential calculus. In Łukasiewicz (1970) (pp. 131–152).Google Scholar
  18. 18.
    Malinowski, G. (1993). Many-valued logics. Oxford: Clarendon Press.Google Scholar
  19. 19.
    McCall, S. (1967). Polish logic: 1920–1939. London, UK: Oxford University Press.Google Scholar
  20. 20.
    Meyer, R.K., & Routley, R. (1972). Algebraic analysis of entailment I. Logique et Analyse, 15, 407–428.Google Scholar
  21. 21.
    Priest, G. (2002). Paraconsistent logic. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 6, pp. 287–393). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  22. 22.
    Robles, G., & Méndez, J.M. (2008). The basic constructive logic for a weak sense of consistency. Journal of Logic Language and Information, 17(1), 89–107.CrossRefGoogle Scholar
  23. 23.
    Robles, G., & Méndez, J.M. A Routley–Meyer semantics for truth-preserving and well-determined Łukasiewicz’s 3-valued logics (Manuscript).Google Scholar
  24. 24.
    Rose, A., & Rosser, J.B. (1958). Fragments of many-valued statement calculi. Transactions of the American Mathematical Society, 87, 1–53.CrossRefGoogle Scholar
  25. 25.
    Routley, R., & Meyer, R.K. (1972). Semantics of entailment III. Journal of Philosophical Logic, 1, 192–208.CrossRefGoogle Scholar
  26. 26.
    Routley, R., & Routley, V. (1972). Semantics of first-degree entailment. Noûs, 1, 335–359.CrossRefGoogle Scholar
  27. 27.
    Routley, R., Meyer, R.K., Plumwood, V., Brady R.T. (1982). Relevant logics and their rivals (Vol. 1). Atascadero, CA: Ridgeview Publishing Co.Google Scholar
  28. 28.
    Slaney, J. (1995). MaGIC, matrix generator for implication connectives: version 2.1, notes and guide. Canberra: Australian National University. URL: http://users.rsise.anu.edu.au/~jks.
  29. 29.
    Suszko, R. (1975). Remarks on Łukasiewicz’s three-valued logics. Bulletin of the Section of Logic, 4, 87–90.Google Scholar
  30. 30.
    Tokarz, M. (1974). A method of axiomatization of Łukasiewicz logics. Studia Logica, 33, 333–338.CrossRefGoogle Scholar
  31. 31.
    Tsuji, M. (1998). Many-valued logic and Suszko thesis revisited. Studia Logica, 60, 299–309.CrossRefGoogle Scholar
  32. 32.
    Tuziak, R. (1988). An axiomatization of the finite-valued Łukasiewicz calculus. Studia Logica, 47, 49–55.CrossRefGoogle Scholar
  33. 33.
    Urquhart, A. (2002). Basic many-valued logic. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 2, pp. 249–295). Dordrecht: Kluwer Academic Publishers.Google Scholar
  34. 34.
    Van Fraasen, B. (1969). Facts and tautological entailments. The Journal of Philosophy, 67, 477–487.CrossRefGoogle Scholar
  35. 35.
    Wajsberg, M. (1931). Axiomatization of the 3-valued propositional calculus. In S. McCall (1967) (pp. 264–284)..Google Scholar
  36. 36.
    Wojcicki, R. (1974). The logics stronger than Łukasiewicz’z three-valued sentential calculus—the notion of degree of maximality versus the notion of degree of completeness. Studia Logica, 33, 201–214.CrossRefGoogle Scholar
  37. 37.
    Wojcicki, R. (1984). Lectures on Propositional Calculi. Ossolineum.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Gemma Robles
    • 1
  • Francisco Salto
    • 1
  • José M. Méndez
    • 2
  1. 1.Dpto. de Psicología, Sociología y FilosofíaUniversidad de LeónLeónSpain
  2. 2.Universidad de SalamancaSalamancaSpain

Personalised recommendations