Admissible orders have brought the structure of chains in the framework of interval-valued fuzzy sets. However, a deeper study of functions monotone with respect to admissible orders is still missing in the literature. In this work, we consider the construction of negations and strong negations on intervals with respect to admissible orders, in particular, for the Xu and Yager and lexicographical orders, as well as for those based on K _α operators. We introduce and discuss an approach to the construction of strong negations on intervals with respect to K _α,β orders based on an arbitrary couple of strong negations defined over the standard real interval [0, 1]. The introduced strong negations have a deep impact on all fields exploiting fuzzy methods dealing with intervals, allowing to introduce complements, dual aggregations, implications, entropies, etc.

In many domains of our daily life (e.g., agriculture, forestry, health, etc.), both laymen and experts need to classify entities into two binary classes (yes/no, good/bad, sufficient/insufficient, benign/malign, etc.). For many entities, this decision is difficult and we need another class called “maybe”, which contains a corresponding quantifiable tendency toward one of these two opposites. Human domain experts are often able to mark any entity, place it in a different class and adjust the position of the slope in the class. Moreover, they can often explain the classification space linguistically—depending on their individual domain experience and previous knowledge. We consider this human-in-the-loop extremely important and call our approach actionable explainable AI. Consequently, the parameters of the functions are adapted to these requirements and the solution is explained to the domain experts accordingly. Specifically, this paper contains three novelties going beyond the state-of-the-art: (1) A novel method for detecting the appropriate parameter range for the averaging function to treat the slope in the “maybe” class, along with a proposal for a better generalisation than the existing solution. (2) the insight that for a given problem, the family of t-norms and t-conorms covering the whole range of nilpotency is suitable because we need a clear “no” or “yes” not only for the borderline cases. Consequently, we adopted the Schweizer–Sklar family of t-norms or t-conorms in ordinal sums. (3) A new fuzzy quasi-dissimilarity function for classification into three classes: Main difference, irrelevant difference and partial difference. We conducted all of our experiments with real-world datasets.