Advertisement

On Standard Completeness for Non-commutative Many-Valued Logics

  • Denisa DiaconescuEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 577)

Abstract

In this paper we present two methods for proving the standard completeness for psMTLr logic: the original one given by Jenei and Montagna [1] and an alternative proof based on Horčik’s method for proving standard completeness theorems. Further, we introduce two extensions of psMTLr logic that still enjoy the standard completeness, i.e. psSMTLr and psIMTLr logics.

Keywords

Non-commutative Logics Many-valued Logics psMTLr Logic Standard Completeness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Jenei, S., Montagna, F.: A proof of standard completeness for non-commutative monoidal t-norm logic. Neural Network World 13, 481–488 (2003)Google Scholar
  2. 2.
    Łukasiewicz, J., Tarski, A.: Untersuchungen über den aussagenkalkül. Comptes Rendus Séances Société dec Sciences et Lettres Varsovie 23, 30–50 (1930)zbMATHGoogle Scholar
  3. 3.
    Hájek, P.: Metamathematics of Fuzzy Logic. Number 4, Dordrecht (1998)Google Scholar
  4. 4.
    Hájek, P., Godo, L., Esteva, F.: A complete many-valued logic with product-conjunction. Arch. Math. Logic 35(3), 191–208 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Esteva, F., Godo, L.: Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124(3), 271–288 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Flondor, P., Georgescu, G., Iorgulescu, A.: Pseudo-t-norms and pseudo-BL algebras. Soft Computing 5, 355–371 (2001)Google Scholar
  7. 7.
    Hájek, P.: Observations on non-commutative fuzzy logic. Soft Computing 8, 38–43 (2003)zbMATHCrossRefGoogle Scholar
  8. 8.
    Leuştean, I.: Non-commutative łukasiewicz propositional logic. Arch. Math. Logic 45, 191–213 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Diaconescu, D.: Non-commutative product logic and probability of fuzzy events. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part II. CCIS, vol. 298, pp. 194–205. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Diaconescu, D.: Non-commutative fuzzy logic psmtl: an alternative proof for the standard completeness theorem. In: Proceedings of the 4th International Conference on Fuzzy Computation Theory and Applications (FCTA 2012), pp. 350–356 (2012), doi:10.5220/0004151603500356Google Scholar
  11. 11.
    Cintula, P., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic. College Publications (2012)Google Scholar
  12. 12.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: An algebraic glimpse at substructural logics. In: Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier (2007)Google Scholar
  13. 13.
    Běhounek, L., Cintula, P.: Fuzzy logics as the logics of chains. Fuzzy Sets and Systems 157(5), 604–610 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kühr, J.: Pseudo-BL-algebras and DRL-monoids. Math. Bohem. 128, 199–208 (2003)Google Scholar
  15. 15.
    Horčik, R.: Alternative proof of standard completeness theorem for mtl. Soft Computing 11(2), 123–129 (2007)zbMATHCrossRefGoogle Scholar
  16. 16.
    Hrbacek, K., Jech, T.: Introduction to set theory. Monographs and textbooks in pure and applied mathematics. Dekker, New York (1999)zbMATHGoogle Scholar
  17. 17.
    Esteva, F., Gispert, J., Godo, L., Montagna, F.: On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Studia Logica 71(2), 199–226 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Jenei, S., Montagna, F.: A proof of standard completeness for esteva and godo’s logic mtl. Studia Logica 70(2), 183–192 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Iorgulescu, A.: Classes of pseudo-bck algebras - part i. J. of Mult.-Valued Logic and Soft Comp. 12(1-2), 71–130 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

Personalised recommendations