Elsevier

Fuzzy Sets and Systems

Volume 76, Issue 3, 22 December 1995, Pages 289-307
Fuzzy Sets and Systems

Measurement-theoretic justification of connectives in fuzzy set theory

https://doi.org/10.1016/0165-0114(95)00067-4Get rights and content

Abstract

The problem of representing intersection and union in fuzzy set theory is considered. There are various proposals in the literature to model these concepts. The possibility of using continuous triangular norms and conorms (including min and max) are taken up in a measurement-theoretic setting. The conditions are laid out to arrive at cardinal scales on which addition and multiplication are meaningful and critically discussed. These conditions must either be accepted on normative grounds or must be empirically verified before the modeling process in order to see which operations are meaningful. It is emphasized that the Archimedean axiom and the existence of natural bounds are crucial in arriving at ratio and absolute scale representations.

References (45)

  • R.R. Yager

    A measurement-informational discussion of fuzzy union and intersection

    Internat. J. Man-Machine Studies

    (1979)
  • R.R. Yager

    On a general class of fuzzy connectives

    Fuzzy Sets and Systems

    (1980)
  • L.A. Zadeh

    Fuzzy sets

    Inform. Control

    (1965)
  • L.A. Zadeh

    A fuzzy-algorithmic approach to the definition of complex or imprecise concepts

    Internat. J. Man-Machine Studies

    (1976)
  • J. Aczél

    Lectures on Functional Equations and Applications

    (1966)
  • P. Bollman-Sdorra et al.

    A measurement-theoretic axiomatization of fuzzy sets

    Fuzzy Sets and Systems

    (1993)
  • A.H. Clifford

    Naturally and totally ordered commutative semigroups

    Amer. J. Math.

    (1954)
  • D. Dubois

    Belief structures, possibility theory and decomposable confidence measures on finite sets

    Comput. Artificial Intelligence

    (1986)
  • D. Dubois

    Possibility theory: searching for normative foundations

  • D. Dubois et al.

    Fuzzy Sets and Systems: Theory and Applications

    (1980)
  • D. Dubois et al.

    A class of fuzzy measures based on triangular norms: a general framework for the combination of information

    Internat. J. General Systems

    (1982)
  • P.C. Fishburn

    Interval Orders and Interval Graphs: a Study of Partially Ordered Sets

    (1985)
  • Cited by (0)

    View full text