Skip to main content
SpringerLink
Log in

Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

  • Original Paper
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with respect to chains defined over the rational unit interval with a special attention to the completeness with respect to the canonical chain, i.e. the algebra over \([0,1] \cap {{\mathbb{Q}}}\) where each truth-constant is interpreted in its corresponding rational truth-value. Finally, we study rational completeness results when we restrict ourselves to deductions between the so-called evaluated formulae.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. See Běhounek and Cintula (2006) for some rationale supporting the idea of fuzzy logics as the logics of chains.

  2. Notice that we could consider parameterized isomorphic t-norms \(*^c_{n_1}\) or \(*^c_{n_2}\) : in the first the case by defining the negation using any \(c \in (0,1)\) instead of \({\frac{1}{2}},\) and in the second case by using any \(c \in ({\frac{1}{2}},1)\) instead of \({\frac{2}{3}}\) (and \(1-c\) instead of \({\frac{1}{3}}\)).

  3. Looking at Table 2, this is the case of the S\({\fancyscript{R}}\)C and the Canonical completeness properties for some of the logics under our scope.

  4. Actually, the result we use from Cignoli (2000) comes from a translation of the paper in Russian (Gurevich and Kokorin 1963).

  5. Notice that the particular choice of an element \(a \in (0,1)^{\mathbb{Q}}\) is not important, as all the resulting algebras are isomorphic and hence they yield the same logic.

References

  • Běhounek L, Cintula P (2006) Fuzzy logics as the logics of chains. Fuzzy Sets Syst 157:604–610

    Article  MATH  Google Scholar 

  • Blok WJ, Pigozzi D (1989) Algebraizable logics. Mem Am Math Soc 396, vol 77

    Google Scholar 

  • Cignoli R (2000) Notas para el curso Álgebras de la lógica borrosa. Department of Mathematics, F.C.E. y N., University of Buenos Aires (in English)

  • Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput 4:106–112

    Article  Google Scholar 

  • Cignoli R, Esteva F, Godo L (2007) On Lukasiewicz logic with truth constants. In: Castillo O et al (eds) Theoretical advances and applications of fuzzy logic and soft computing ASC 42. Springer, Heidelberg, pp 869–875

  • Cintula P, Esteva F, Gispert J, Godo L, Montagna F, Noguera C (2009) Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann Pure Appl Logic (to appear)

  • Esteva F, Domingo X (1980) Sobre negaciones fuertes y débiles en [0,1]. Stochastica 4:141–166

    MATH  MathSciNet  Google Scholar 

  • Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124:271–288

    Article  MATH  MathSciNet  Google Scholar 

  • Esteva F, Gispert J, Godo L, Montagna F (2002) On the standard and rational completeness of some axiomatic extensions of monoidal t-norm based logic. Stud Log 71:199–226

    Article  MATH  MathSciNet  Google Scholar 

  • Esteva F, Godo L, Montagna F (2004) Equational characterization of the subvarieties of BL generated by t-norm algebras. Stud Log 76:161–200

    Article  MATH  MathSciNet  Google Scholar 

  • Esteva F, Godo L, Noguera C (2006a) On rational weak nilpotent minimum logics. J Multiple Valued Log Soft Comput 12:9–32

    MATH  MathSciNet  Google Scholar 

  • Esteva F, Godo L, Noguera C (2006b) Real, rational and finite chain semantics for fuzzy logics, Actas del XIII congreso español sobre tecnologí as y lógica fuzzy ESTYLF 2006. Ciudad Real, pp 83–88

  • Esteva F, Gispert J, Godo L, Noguera C (2007a) Adding truth-constants to logics of a continuous t-norm: axiomatization and completeness results. Fuzzy Sets Syst 158:597–618

    Article  MATH  MathSciNet  Google Scholar 

  • Esteva F, Godo L, Noguera C (2007b) On completeness results for predicate Łukasiewicz, Product, Gödel, and Nilpotent minimum logics expanded with truth-constants. Mathw Soft Comput 14:233–246

    MATH  MathSciNet  Google Scholar 

  • Esteva F, Godo L, Noguera C (2009) On expansions of t-norm based logics with truth-constants. In: Gottwald S, Hájek P, Höhle U, Klement EP (eds) Fuzzy logics and related structures (to appear)

  • Gurevich YS, Kokorin AI (1963) Universal equivalence of ordered abelian groups. Algebra Log 2:37–39 (in Russian)

    MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic. Trends in logic-studia logica library, vol 4. Kluwer, Dordrecht

  • Hájek P (2006) Computational complexity of t-norm based propositional fuzzy logics with rational truth-constants. Fuzzy Sets Syst 157:677–682

    Article  MATH  Google Scholar 

  • Jenei S, Montagna F (2002) A proof of standard completeness for Esteva and Godo’s logic MTL. Stud Log 70:183–192

    Article  MATH  MathSciNet  Google Scholar 

  • Ling CH (1965) Representation of associative functions. Publ Math Debrecen 12:189–212

    MathSciNet  Google Scholar 

  • Mostert PS, Shields AL (1957) On the structure of semigroups on a compact manifold with boundary. Ann Math 65:117–143

    Article  MathSciNet  Google Scholar 

  • Novák V (1990a) On the syntactico-semantical completeness of first-order fuzzy logic. Part I. Syntax and semantics. Kybernetika 26:47–66

    MATH  MathSciNet  Google Scholar 

  • Novák V (1990b) On the syntactico-semantical completeness of first-order fuzzy logic. Part II. Main results. Kybernetika 26:134–154

    MATH  MathSciNet  Google Scholar 

  • Noguera C, Esteva F, Gispert J (2008) On triangular norm based axiomatic extensions of the weak Nilpotent minimum logic. Math Log Q 54:387–409

    Article  MATH  MathSciNet  Google Scholar 

  • Pavelka J (1979) On fuzzy logic I, II, III. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 25:45–52, 119–134, 447–464

  • Savický P, Cignoli R, Esteva F, Godo L, Noguera C (2006) On product logic with truth-constants. J Log Comput 16:205–225

    Article  MATH  Google Scholar 

  • Turunen E (1999) Mathematics behind fuzzy logic. Springer, Heidelberg

    MATH  Google Scholar 

Download references

Acknowledgments

The authors acknowledge partial support by the Spanish project MULOG2 (TIN2007-68005-C04), including feder funds of the European Union, and by the Generalitat de Catalunya grant 2005-SGR-00093. The third author also acknowledges partial support from the grant 2006-BP-A-10043 of the Departament d’Educació i Universitats of the Generalitat de Catalunya.

Author information

Authors and Affiliations

  1. IIIA, CSIC, Campus de la Universitat Autònoma de Barcelona s/n, 08193, Bellaterra, Catalonia, Spain

    Francesc Esteva & Lluís Godo

  2. Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Carles Noguera

Authors
  1. Francesc Esteva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lluís Godo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carles Noguera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesc Esteva.

Additional information

This is an extended and revised version of the paper Rational Completeness Results for Prominent Propositional Fuzzy Logics with Truth-Constants. Proc. of ESTYLF’08, Mieres, Spain, pp. 133–139.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Esteva, F., Godo, L. & Noguera, C. Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics. Soft Comput 14, 273–284 (2010). https://doi.org/10.1007/s00500-009-0402-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-009-0402-8

Keywords

Search

Navigation