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 P_i (such as ldquohigh feverrdquo, ldquoheadacherdquo, etc.), we know the exact set S_i of all the objects that satisfy this property. In practice, we usually only have partial knowledge. In this case, we only know the set S_i of all the objects about which we know that P_i holds and the set S_i about which we know that P_i may hold (i.e., equivalently, that we have not yet excluded the possibility of P_i). 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 S_i 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 S_i. In this paper, we describe how to compute the class S of all possible sets S.