Fuzzy cardinality and the modeling of imprecise quantification.

*(English)*Zbl 0601.03006This paper proposes a systematic investigation of the concept of cardinality of a fuzzy set and tries to unify two kinds of definitions of the cardinality of fuzzy sets proposed by several authors by putting them into a single setting. The insufficiencies of the usual representation of a fuzzy set by means of its level-cuts are shown and a new proposal is made. It is indicated that several basic properties of cardinality are preserved. Scalar and fuzzy cardinalities are studied through the notion of upper and lower expectations. The scalar cardinality proves to be the upper mean value of the fuzzy cardinality. The question of the probability of a fuzzy event is dealt with and most results which hold for cardinality are extended to probability in a finite setting. Examples of imprecise quantification modelling in simple natural language statements are discussed and it is stressed that, while most existing meaning-representation procedures rely on scalar cardinality, it is possible to use fuzzy cardinality for the same purpose.

Reviewer: M.Mizumoto

##### MSC:

03B52 | Fuzzy logic; logic of vagueness |

03B65 | Logic of natural languages |

03E72 | Theory of fuzzy sets, etc. |

##### Keywords:

possibility; linguistic quantifiers; cardinality of a fuzzy set; probability of a fuzzy event; imprecise quantification modelling; natural language statements
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\textit{D. Dubois} and \textit{H. Prade}, Fuzzy Sets Syst. 16, 199--230 (1985; Zbl 0601.03006)

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