Sinha-Dougherty approach to the fuzzification of set inclusion revisited

Inclusion for fuzzy sets was &rst introduced by Zadeh in his seminal 1965 paper. Since it was found that the de&nition of inclusion was not in the true spirit of fuzzy logic, various researchers have set out to de&ne alternative indicators of the inclusion of one fuzzy set into another. Among these alternatives, the indicator proposed by Sinha and Dougherty stands out as an intuitively appealing one, as it is built up with a strong but appropriate collection of axioms in mind. Starting from a very general expression depending on four functional parameters for such an indicator, those authors proposed conditions they claimed to be necessary and su7cient to satisfy the axioms. This paper aims to revisit this material by exposing it in a clearer way, correcting errors along the way while pinpointing some nasty pitfalls that Sinha and Dougherty overlooked. This results in a new, easier to handle and more consistent framework for the axiomatic characterization of inclusion grades for fuzzy sets, advantageous to the further development of practical applications. In the end, a link is established with Kitainik’s results on the fuzzi&cation of set inclusion, allowing amongst others the derivation of a su7cient and necessary characterization of the Sinha–Dougherty axioms. c © 2002 Elsevier Science B.V. All rights reserved.