Bell-Shaped Inequalities in Hexa-parametric Family of Similarity Measurement

Abstract

In this thesis, we study the T-transitive family of cardinality-based fuzzy similarity andinclusionmeasuresandobtainthenecessaryandsufficientparametricconditionsto characterize the T-transitive members of the parametric family of the cardinality based similarity measure. Firstly, In crisp logic every object is similar to itself with degree of reflexivity 1, while the degree of reflexivity in fuzzy logic can be any value in the unit interval, (0,1]. This behaviour of a fuzzy set is used to enlighten the concept of similarity and inclusion measures and discovering the relations between the parameters of the transitive members of a family of cardinality-based fuzzy measure. We present meta-theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality. The conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection. In particular, this conjunctor should fulfil a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations can be avoided. This is illustrated in the demonstration of the Łukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the Jaccard coefficient, or equivalently, the triangle inequality of the corresponding dissimilarity measures. Finally,anewclassofrationalcardinality-basedsimilaritymeasureusingmultiplication of cardinalities is introduced and discussed the monotonicity, boundary conditions and transitivity.