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Negation and affirmation: the role of involutive negators

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Abstract

Strict negators and automorphisms are prevalently used to fuzzify the Boolean negation and affirmation. Involutive negators are of particular interest. Every monotone [0,1] → [0,1] bijection is a composition of at most four involutive negators. Involutive negators are geometrically recognized by the symmetry of their graph w.r.t. the first bisector. If the graph of an automorphism has an alternating behavior, we can generate the automorphism by a pair of involutive negators.

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References

  • Batyrshin I (2003) On the structure of involutive, contracting and expanding negations. Fuzzy Sets Syst 139:661–672

    Article  MATH  MathSciNet  Google Scholar 

  • Batyrshin I, Wagenknecht M (1997) Noninvolutive negations on [0,1]. J Fuzzy Math 5:997–1010

    MATH  MathSciNet  Google Scholar 

  • Batyrshin I, Wagenknecht W (1998) Contracting and expanding negations on [0,1]. J Fuzzy Math 6:133–140

    MATH  MathSciNet  Google Scholar 

  • Calvo T, Kolesárová A, Komorníková M, Mesiar R (2002) Aggregation operators: properties, classes and construction methods, ser. studies in fuzziness and soft computing, vol 97. Physica-Verlag, Heidelberg, pp 3–104

  • Choquet G (1954) Theory of capacities. Annales de l’Institut Fourier 5:131–295

    MathSciNet  Google Scholar 

  • De Baets B (1999) Idempotent uninorms. Eur J Oper Res 118:631–642

    Article  MATH  Google Scholar 

  • De Cooman G, Kerre EE (1994) Order norms on bounded partially ordered sets. J Fuzzy Math 2:281–310

    MATH  MathSciNet  Google Scholar 

  • Dubois D, Prade H (1988) Possibility theory: an approach to computerised processing of uncertainty. Plenum Press, New York

    Google Scholar 

  • Esteva F, Domingo X (1980) Sobre funciones de negación en [0,1]. Stochastica 4:141–166

    MATH  MathSciNet  Google Scholar 

  • Fodor J (1993) A new look at fuzzy connectives. Fuzzy Sets Syst 57:141–148

    Article  MATH  MathSciNet  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic, ser. trends in logic, vol 4. Kluwer, Dordrecht

    Google Scholar 

  • Higashi M, Klir G (1982) On measures of fuzziness and fuzzy complements. Int J Gen Syst 8:169–180

    MATH  MathSciNet  Google Scholar 

  • Klement E-P, Mesiar R, Pap E (2000) Triangular norms, ser. trends in logic, vol 8. Kluwer, Dordrecht

    Google Scholar 

  • Maes K, De Baets B (2005) Isomorphic continuous connectives in fuzzy logic. J Intell Fuzzy Syst 16:273–279

    Google Scholar 

  • Maes KC, De Baets B (2006) Orthosymmetrical monotone functions. B Belg Math Soc Sim (in press)

  • Mayor G (1994) Sugeno’s negations and t-norms. Mathware Soft Comput 1:93–98

    MATH  MathSciNet  Google Scholar 

  • Mayor G, Calvo T (1994) Fractal negations. Mathware Soft Comput 3:277–283

    MathSciNet  Google Scholar 

  • Mesiar R, Rückschlossová T (2004) Characterization of invariant aggregation operators. Fuzzy Sets Syst 142:63–73

    Article  MATH  Google Scholar 

  • Nguyen H, Walker E (2000) A first course in fuzzy logic, 2nd edn. Chapman & Hall/CRC, London

    MATH  Google Scholar 

  • Ovchinnikov S (1983) General negations in fuzzy set theory. J Math Anal Appl 92:234–239

    Article  MATH  MathSciNet  Google Scholar 

  • Ovchinnikov S, Dukhovny A (2000) Integral representation of invariant functionals. J Math Anal Appl 244:228–232

    Article  MATH  MathSciNet  Google Scholar 

  • Trillas E (1979) Sobre funciones de negación en la teoría de conjuntos difusos. Stochastica 3:47–59

    MATH  MathSciNet  Google Scholar 

  • Trillas E, Rodríguez A, Cubillo S (1994) Ovchinnikov’s automorphisms revisited. Mathware Soft Comput 1:83–92

    MATH  MathSciNet  Google Scholar 

  • Wagenknecht M, Batyrshin I (1998a) On negations generated by compensations. J Fuzzy Math 6:141–150

    MATH  MathSciNet  Google Scholar 

  • Wagenknecht M, Batyrshin I (1998b) Fixed point properties of fuzzy negations. J Fuzzy Math 6:975–981

    MATH  MathSciNet  Google Scholar 

  • Wangming W (1994) Commutative implications on complete lattices. Int J Uncertainty Fuzziness Knowl Based Syst 2:333–341

    Article  Google Scholar 

  • Yager R (1980) On measures of fuzziness and negation. II. Lattices. Inform Control 44:236–260

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, 9000, Gent, Belgium

    Koen C. Maes & Bernard De Baets

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  1. Koen C. Maes
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  2. Bernard De Baets
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Correspondence to Koen C. Maes.

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Maes, K.C., De Baets, B. Negation and affirmation: the role of involutive negators. Soft Comput 11, 647–654 (2007). https://doi.org/10.1007/s00500-006-0127-x

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