Saskia Janssens

Inequalities in fuzzy probability calculus

We present all Bell-type inequalities concerning at most four random events of which not more than two are intersected at the same time. Reformulating these inequalities in the context of fuzzy probability calculus leads to related inequalities on commutative conjunctors, and in particular on triangular norms and commutative copulas. For the most important parametric families of t-norms, we identify the parameter values for which each of the inequalities is fulfilled. Some of the inequalities hold for any commutative copula, while all of them are preserved under ordinal sums.

Transitivity of comparison measures

A family of fuzzification schemes is proposed for transforming cardinality-based similarity and inclusion measures for ordinary sets into similarity and inclusion measures for fuzzy sets in a finite universe. The family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets. In particular, the fuzzy set intersections are pointwisely generated by Frank t-norms. The fuzzification schemes are applied to a variety of previously studied rational cardinality-based similarity and inclusion measures for ordinary sets and it is demonstrated that transitivity is preserved in the fuzzification process. Finally, we introduce a new parametrized family of similarity measures and a new parametrized family of inclusion measures and investigate the transitivity properties of both families and their fuzzification.

Identities and Inequalities on Fuzzy Set Cardinalities

In this paper, we study identities and inequalities on fuzzy set cardinalities. An important such identity is the valuation property. Other identities are related to the fuzzification of the (symmetric) difference of fuzzy sets. We show that using a stable, commutative quasi-copula and its dual operator for modelling intersection and union of fuzzy sets, identities which are valid in the crisp case, also hold for cardinalities of fuzzy sets. The basic inequalities that come to mind are the Bell inequalities. We show that our results on the fuzzified Bell inequalities can be exploited to develop a framework in which the validity of more general inequalities on fuzzy cardinalities can be checked easily. An interesting application of these meta-theorems can be found in the field of similarity measurement