A model for the measurement of membership and the consequences of its empirical implementation.

*(English)*Zbl 0538.94026This research regards a controversial problem in the fuzzy set theory: the measurement of fuzziness; here a measurement model is proposed when the domain of discourse is order-dense (has an associated physical continuum). The authors prove - by representation and uniqueness theorems - that, in the above conditions, the membership of a fuzzy set is on an interval scale; the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership are the arguments in the support of the measurement model. The preliminary results of an empirical study for the verification of this model and the construction of membership functions are also presented. Discussing the meaningfulness of operations on membership, the authors propose a method which involves the replacement of the membership function by a function derived from it, but which is on an absolute scale.

Reviewer: L.Olaru

##### MSC:

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

91E99 | Mathematical psychology |

##### Keywords:

scale invariance; fuzzy set; measurement of fuzziness; interval scale; empirical study; membership functions; absolute scale
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\textit{A. M. Norwich} and \textit{I. B. Turksen}, Fuzzy Sets Syst. 12, 1--25 (1984; Zbl 0538.94026)

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