Towards More Adequate Representation of Uncertainty: From Intervals to Set Intervals, with the Possible Addition of Probabilities and Certainty Degrees

In the ideal case of complete knowledge, for each property Pi (such as high fever , headache , etc.), we know the exact set Si of all the objects that satisfy this property. In practice, we usually only have partial knowledge. In this case, we only know the set Si- of all the objects about which we know that Pi holds and the set Si+ about which we know that Pi may hold (i.e., equivalently, that we have not yet excluded the possibility of Pi). This pair of sets is called a set interval. Based on the knowledge of the original properties, we would like to describe the set S of all the values that satisfy some combination of the original properties: e.g., high fever and headache and not rash. In the ideal case when we know the exact set Si of all the objects satisfying each property, it is sufficient to apply the corresponding set operation (composition of union, intersection, and complement) to the known sets Si. In this paper, we describe how to compute the class [S] of all possible sets S