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Archive for Mathematical Logic

, Volume 50, Issue 3–4, pp 257–285 | Cite as

n-Contractive BL-logics

  • Matteo Bianchi
  • Franco Montagna
Original Article

Abstract

In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in Hájek (Metamathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom \({\phi^n\rightarrow\phi^{n+1}}\), for some \({n\in\mathbb{N}^+}\)) axiomatic extensions of BL and their corresponding varieties: BL n , SBL n , BL n and SBL n . Concerning BL n we have that every BL n -chain is isomorphic to an ordinal sum of MV-chains of at most n + 1 elements, whilst every BL n -chain is isomorphic to an ordinal sum of MV n -chains (for SBL n and SBL n a similar property holds, with the difference that the first component must be the two elements boolean algebra); all these varieties are locally finite. Moving to the content of the paper, after a preliminary section, we will study generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we will analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.

Keywords

Many-Valued logics Basic logic n-Contractive logics Residuated lattices Varieties of lattices MV-algebras 

Mathematics Subject Classification (2000)

03B47 03B50 03C07 03G10 06B20 06D35 

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References

  1. 1.
    Aglianò P., Ferreirim I., Montagna F.: Basic Hoops: an Algebraic Study of Continuous t-norms. Studia Logica 87(1), 73–98 (2007). doi: 10.1007/s11225-007-9078-1 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aglianò P., Montagna F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181(2–3), 105–129 (2003). doi: 10.1016/S0022-4049(02)00329-8 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Aguzzoli, S., Bova, S., Marra, V.: Applications of finite duality to locally finite varieties of BL-algebras. In: Artemov, S., Nerode, A. (eds.), Logical Foundations of Computer Science—International Symposium, LFCS 2009, Deerfield Beach, FL, USA, January 3–6, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5407, pp. 1–15. Springer, Berlin (2009). doi: 10.1007/978-3-540-92687-0_1
  4. 4.
    Bianchi M., Montagna F.: Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Log. 48(8), 719–736 (2009). doi: 10.1007/s00153-009-0145-3 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Blok W., Ferreirim I.: On the structure of hoops. Algebra Universalis 43(2–3), 233–257 (2000). doi: 10.1007/s000120050156 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Blok, W., Pigozzi, D.: Algebraizable logics. No. 396 in memoirs of The American Mathematical Society. Am. Math. Soc. (1989); ISBN:0-8218-2459-7. Available on http://orion.math.iastate.edu/dpigozzi/
  7. 7.
    Burris, S., Sankappanavar, H.P.: A course in Universal Algebra, Graduate Texts in Mathematics, vol. 78. Springer, New York (1981). An updated and revised electronic edition is available on http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
  8. 8.
    Busaniche M., Cabrer L.: Canonicity in subvarieties of BL-algebras. Algebra Universalis 62(4), 375–397 (2009). doi: 10.1007/s00012-010-0055-6 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ciabattoni A., Esteva F., Godo L.: T-norm based logics with n-contraction. Neural Netw. World 16(5), 453–495 (2008). doi: 10.1093/jigpal/jzn014 Google Scholar
  10. 10.
    Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning, Trends in Logic, vol. 7. Kluwer Academic Publishers, Berlin (1999); ISBN:9780792360094Google Scholar
  11. 11.
    Cintula, P.: From fuzzy logic to fuzzy mathematics. Ph.D. thesis, FNSPE CTU, Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering—Czech Technical University in Prague, Trojanova 13, 120 00 Prague 2, Czech Republic (2004). Available on http://www2.cs.cas.cz/~cintula/thesis.pdf
  12. 12.
    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. Log. 160(1), 53–81 (2009). doi: 10.1016/j.apal.2009.01.012 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Cintula P., Hájek P.: Triangular norm predicate fuzzy logics. Fuzzy Sets Syst. 161(3), 311–346 (2010). doi: 10.1016/j.fss.2009.09.006 CrossRefzbMATHGoogle Scholar
  14. 14.
    Davey B.A., Priestley H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002). doi: 10.2277/0521784514 ISBN:9780521784511zbMATHGoogle Scholar
  15. 15.
    di Nola A., Lettieri A.: One chain generated varieties of MV-algebras. J. Algebra 225(2), 667–697 (2000). doi: 10.1006/jabr.1999.8136 CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001). doi: 10.1016/S0165-0114(01)00098-7 CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Ferreirim, I.: On varieties and quasivarieties of hoops and their reducts. Ph.D. thesis, University of Illinois at Chicago, Chicago (1992)Google Scholar
  18. 18.
    Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: an Algebraic Glimpse at Substructural Logics, Studies in Logic and The Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007) ISBN:978-0-444-52141-5Google Scholar
  19. 19.
    Grigolia, R.: Algebraic analysis of Łukasiewicz-tarski n-valued logical systems. In: Selected Papers on Łukasiewicz Sentencial Calculi, pp. 81–91. Polish Academy of Science, Ossolineum (1977)Google Scholar
  20. 20.
    Hájek P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4, paperback edn. Kluwer Academic Publishers, Berlin (1998) ISBN:9781402003707Google Scholar
  21. 21.
    Horčík R., Noguera C., Petrík M.: On n-contractive fuzzy logics. Math. Log. Quart. 53(3), 268–288 (2007). doi: 10.1002/malq.200610044 CrossRefzbMATHGoogle Scholar
  22. 22.
    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, 2008. Samui, Thailand (2008)Google Scholar
  23. 23.
    MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42(3), 416–460 (1937). Available on http://www.jstor.org/stable/1989739
  24. 24.
    Metcalfe G., Montagna F.: Substructural fuzzy logics. J. Symbolic Log. 72(3), 834–864 (2007). doi: 10.2178/jsl/1191333844 CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Montagna F.: Interpolation and Beth’s property in propositional many-valued logics: a semantic investigation. Ann. Pure. Appl. Log. 141(1–2), 148–179 (2006). doi: 10.1016/j.apal.2005.11.001 CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Montagna F., Noguera C.: Arithmetical complexity of first-order predicate fuzzy logics over distinguished semantics. J. Log. Comput. 20(2), 399–424 (2010). doi: 10.1093/logcom/exp052 CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Montagna F., Noguera C., Horčík R.: On weakly cancellative fuzzy logics. J. Log. Comput. 16(4), 423–450 (2006). doi: 10.1093/logcom/exl002 CrossRefzbMATHGoogle Scholar
  28. 28.
    van Alten, C.J.: Preservation theorems for MTL-Chains. Log. J. IGPL (2010). doi: 10.1093/jigpal/jzp088

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

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

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