Notre Dame Journal of Formal Logic

Structural Completeness in Fuzzy Logics

Petr Cintula and George Metcalfe

Abstract

Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.

Article information

Source
Notre Dame J. Formal Logic Volume 50, Number 2 (2009), 153-182.

Dates
First available in Project Euclid: 11 May 2009

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1242067708

Digital Object Identifier
doi:10.1215/00294527-2009-004

Mathematical Reviews number (MathSciNet)
MR2535582

Zentralblatt MATH identifier
1190.03027

Subjects
Primary: 03B22: Abstract deductive systems 03B52: Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05] 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

Keywords
substructural logics fuzzy logics structural completeness admissible rules primitive variety residuated lattices

Citation

Cintula, Petr; Metcalfe, George. Structural Completeness in Fuzzy Logics. Notre Dame J. Formal Logic 50 (2009), no. 2, 153--182. doi:10.1215/00294527-2009-004. http://projecteuclid.org/euclid.ndjfl/1242067708.


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References

  • [1] Aglianò, P., I. M. A. Ferreirim, and F. Montagna, "Basic hoops: An algebraic study of continuous t-norms", Studia Logica, vol. 87 (2007), pp. 73--98.
  • [2] Agliano, P., and F. Montagna, "Varieties of BL"-algebras. I. General properties, Journal of Pure and Applied Algebra, vol. 181 (2003), pp. 105--29.
  • [3] Aguzzoli, S., and S. Bova, "The free n-generated BL"-algebra, forthcoming in Annals of Pure and Applied Logic, (2009).
  • [4] Bergman, C., "Structural completeness in algebra and logic", pp. 59--73 in Algebraic Logic (Budapest, 1988), edited by H. Andréka, J. D. Monk, and I. Nemeti, vol. 54 of Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 1991.
  • [5] Blok, W. J., and I. M. A. Ferreirim, "On the structure of hoops", Algebra Universalis, vol. 43 (2000), pp. 233--57.
  • [6] Blok, W. J., and D. Pigozzi, "Algebraizable logics", Memoirs of the American Mathematical Society, vol. 77 (1989).
  • [7] Blok, W. J., and J. G. Raftery, "Varieties of commutative residuated integral pomonoids and their residuation subreducts", Journal of Algebra, vol. 190 (1997), pp. 280--328.
  • [8] Blount, K., and C. Tsinakis, "The structure of residuated lattices", International Journal of Algebra and Computation, vol. 13 (2003), pp. 437--61.
  • [9] Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1981.
  • [10] Cintula, P., F. Esteva, J. Gispert, L. Godo, F. Montagna, and C. Noguera, "Distinguished algebraic semantics for t-norm based fuzzy logics: Methods and algebraic equivalencies", forthcoming in Annals of Pure and Applied Logic, (2009).
  • [11] Czelakowski, J., Protoalgebraic Logics, vol. 10 of Trends in Logic---Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.
  • [12] Dzik, W., "Unification in some substructural logics of BL"-algebras and hoops, Reports on Mathematical Logic, (2008), pp. 73--83.
  • [13] Dzik, W., and A. Wroński, "Structural completeness of Gödel's and Dummett's propositional calculi", Studia Logica, vol. 32 (1973), pp. 69--73.
  • [14] Esteva, F., and L. Godo, "Monoidal t-norm based logic: Towards a logic for left-continuous t-norms", Fuzzy Sets and Systems, vol. 124 (2001), pp. 271--88.
  • [15] Esteva, F., L. Godo, P. Hájek, and F. Montagna, "Hoops and fuzzy logic", Journal of Logic and Computation, vol. 13 (2003), pp. 531--55.
  • [16] Ferreirim, I. M. A., On Varieties and Quasivarieties of Hoops and Their Reducts, Ph.D. thesis, University of Illinois at Chicago, Chicago, 1992.
  • [17] Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 2007.
  • [18] Hájek, P., Metamathematics of Fuzzy Logic, vol. 4 of Trends in Logic---Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 1998.
  • [19] Makinson, D., "A characterization of structural completeness of a structural consequence operation", Reports on Mathematical Logic, (1976), pp. 99--101.
  • [20] Metcalfe, G., and F. Montagna, "Substructural fuzzy logics", The Journal of Symbolic Logic, vol. 72 (2007), pp. 834--64.
  • [21] Montagna, F., "Generating the variety of BL"-algebras, Soft Computing, vol. 9 (2005), pp. 869--74.
  • [22] Olson, J. S., J. G. Raftery, and C. J. van Alten, "Structural completeness in substructural logics", Logic Journal of the Interest Group of Pure and Applied Logic, vol. 16 (2008), pp. 455--95.
  • [23] Pogorzelski, W. A., "Structural completeness of the propositional calculus", Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 19 (1971), pp. 349--51.
  • [24] Prucnal, T., "On the structural completeness of some pure implicational propositional calculi", Studia Logica, vol. 30 (1972), pp. 45--50.
  • [25] Prucnal, T., and A. Wroński, "An algebraic characterization of the notion of structural completeness", Bulletin of the Section of Logic, vol. 3 (1974), pp. 30--33.
  • [26] Rybakov, V. V., Admissibility of Logical Inference Rules, vol. 136 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1997.
  • [27] Tokarz, M., "On structural completeness of Łukasiewicz's logics", Studia Logica, vol. 30 (1972), pp. 53--57.
  • [28] Wójcicki, R., Theory of Logical Calculi. Basic Theory of Consequence Operations, vol. 199 of Synthese Library, Kluwer Academic Publishers, Dordrecht, 1988.
  • [29] Wojtylak, P., "A new proof of structural completeness of Łukasiewicz's logics", Bulletin of the Section of Logic, vol. 5 (1976), pp. 145--52.
  • [30] Wojtylak, P., "On structural completeness of the infinite-valued Łukasiewicz's propositional calculus", Bulletin of the Section of Logic, vol. 5 (1976), pp. 153--57.
  • [31] Wojtylak, P., "On structural completeness of many-valued logics", Studia Logica, vol. 37 (1978), pp. 139--47.
  • [32] Wroński, A., "On factoring by compact congruences in algebras of certain varieties related to the intuitionistic logic", Bulletin of the Section of Logic, vol. 15 (1986), pp. 48--51.