Supersound many-valued logics and Dedekind-MacNeille completions

  • Matteo Bianchi
  • Franco Montagna


In Hájek et al. (J Symb Logic 65(2):669–682, 2000) the authors introduce the concept of supersound logic, proving that first-order Gödel logic enjoys this property, whilst first-order Łukasiewicz and product logics do not; in Hájek and Shepherdson (Ann Pure Appl Logic 109(1–2):65–69, 2001) this result is improved showing that, among the logics given by continuous t-norms, Gödel logic is the only one that is supersound. In this paper we will generalize the previous results. Two conditions will be presented: the first one implies the supersoundness and the second one non-supersoundness. To develop these results we will use, between the other machineries, the techniques of completions of MTL-chains developed in Labuschagne and van Alten (Proceedings of the ninth international conference on intelligent technologies, 2008) and van Alten (2009). We list some of the main results. The first-order versions of MTL, SMTL, IMTL, WNM, NM, RDP are supersound; the first-order version of an axiomatic extension of BL is supersound if and only it is n-potent (i.e. it proves the formula \({\varphi^{n}\,\to\,\varphi^{n\,{+}\,1}}\) for some \({n\,\in\,\mathbb{N}^+}\)). Concerning the negative results, we have that the first-order versions of ΠMTL, WCMTL and of each non-n-potent axiomatic extension of BL are not supersound.


Many-valued logics Basic properties of first-order languages and structures Lattices and related structures Complete lattices Completions 

Mathematics Subject Classification (2000)

03B50 03C07 03G10 06B23 


  1. 1.
    Aglianó P., Montagna F. (2003). Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181(2–3): 105–129 doi: 10.1016/S0022-4049(02)00329-8 11Google Scholar
  2. 2.
    Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958). 8Google Scholar
  3. 3.
    Cignoli, R., Torrens, A.: An algebraic analysis of product logic. Mult. Valued Logic 5(1), 45–65 (2000) 8Google Scholar
  4. 4.
    Cintula, P., Esteva, F., Gispert, J., Godo, L., Montagna, F., Noguera, C.: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure. Appl. Logic 160(1), 53–81 (2009). doi: 10.1016/j.apal.2009.01.012 5, 17Google Scholar
  5. 5.
    Cintula, P., Hájek, P.: Triangular norm predicate fuzzy logics. Submitted for Publication (2009). A preprint is available on 4, 5
  6. 6.
    Dummett, M.: A propositional calculus with denumerable matrix. J. Symb. Logic 24(2), 97–106 (1959). 2Google Scholar
  7. 7.
    Esteva, F., Gispert, J., Godo, L., Montagna, F.: On the standard and rational completeness of some axiomatic extensions of the monoidal T-norm logic. Stud. Logica 71(2), 199–226 (2002). doi: 10.1023/A:1016548805869 2, 6Google Scholar
  8. 8.
    Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Syst. 124(3), 271–288 (2001). doi: 10.1016/S0165-0114(01)00098-7 1, 2, 3, 6, 7Google Scholar
  9. 9.
    Esteva, F., Godo, L., Hájek, P.: A complete many-valued logics with product-conjunction. Arch. Math. Logic 35(3), 191–208 (1996). doi: 10.1007/BF01268618 2Google Scholar
  10. 10.
    Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Logic 4(2), 103–124 (2000). doi: 10.1007/s001530050006 2Google Scholar
  11. 11.
    Gödel, K.: On the intuitionistic propositional calculus (1932). In: Feferman, S., Jr. Connolly, J.W.D., Goldfarb, W., Parsons, C., Sieg, W. (eds.) Kurt Gödel Collected Works, vol. 1, Publications: 1929–1936. Paperback edn., Oxford University Press, Oxford (2001). ISBN:9780195147209 2Google Scholar
  12. 12.
    Hájek P.: Metamathematics of Fuzzy logic. In: Trends in Logic, vol. 4. Paperback edn., Kluwer Academic Publishers (2002). ISBN: 9781402003707 1, 2, 4, 6, 7, 8, 10Google Scholar
  13. 13.
    Hájek, P.: Observations on the monoidal t-norm logic. Fuzzy Syst. 132(1), 107–112 (2002). doi: 10.1016/S0165-0114(02)00057-X 2Google Scholar
  14. 14.
    Hájek, P., Paris, J.B., Shepherdson, J.C.: Rational pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic. J. Symb. Logic 65(2), 669–682 (2000). 1, 5, 7, 17Google Scholar
  15. 15.
    Hájek, P., Shepherdson, J.C.: A note on the notion of truth in fuzzy logic. Ann. Pure Appl. Logic 109(1-2), 65–69 (2001). doi: 10.1016/S0168-0072(01)00041-0 1, 5, 7, 17Google Scholar
  16. 16.
    Horčík, R., Montagna, F., Noguera, C.: On weakly cancellative Fuzzy logics. J. Logic Comput. 16(4), 423–450 (2006). doi: 10.1093/logcom/exl002 3Google Scholar
  17. 17.
    Łukasiewicz, J., Tarski, A. : Investigations into the sentential calculus. In: Borkowski, L. (ed.) Jan Łukasiewicz Selected Works Studies in Logic and the Foundations of Mathematics, pp. . North Holland Publishing–Amsterdam Polish Scientific, Warszawa (1970). ISBN:720422523 2Google Scholar
  18. 18.
    Labuschagne, C., van Alten, C.: On the MacNeille completion of MTL-chains. In: Proceedings of the Ninth International Conference on Intelligent Technologies, October 7–9, Samui, Thailand (2008) 1, 6, 7, 9Google Scholar
  19. 19.
    Montagna, F., Ono, H.: Kripke semantics, undecidability and standard completeness for Esteva and Godo’s logic MTL∀. Stud. Logica 71(2), 227–245 (2002). doi: 10.1023/A:1016500922708 6
  20. 20.
    Montagna, F., Sacchetti, L.: Kripke-style semantics for many-valued logics. Math. Logic Q. 49(6), 629–641 (2003). doi: 10.1002/malq.200310068 6
  21. 21.
    Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC (2006). Available on 2
  22. 22.
    van Alten, C.J.: Preservation theorems for MTL-chains. Submitted for publication (2009) 1, 6, 7, 9Google Scholar
  23. 23.
    Wang, S.: A Fuzzy logic for the revised drastic product t-norm. Soft Comput. A Fus. Found. Methodol. Appl. 11(6), 585–590 (2007). doi: 10.1007/s00500-006-0134-y 3, 6

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanItaly
  2. 2.Dipartimento di Scienze Matematiche e Informatiche “Roberto Magari”Università degli Studi di SienaSienaItaly

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