Ordering of type-2 values: Representation theorems

In the traditional (2-valued) logic, we assume that each statement is either true or false. In practice, for some statements, we do not know whether they are true or false. It is therefore natural to consider different degrees of confidence; the (partially) ordered set V of all such degrees forms a fuzzy logic. For example, in the standard [0,1]-based fuzzy logic, these degrees form the interval [0,1]. In practice, it is difficult for an expert to describe his or her degree of confidence in a statement by an exact number from the interval [0,1], or, more generally, by an exact element of the corresponding fuzzy logic. At best, an expert can provide a set S sube V of possible values: e.g., a subinterval of the interval [0,1]. For such sets, it is natural to define a relation "possibly more confident" S₁ diam les S₂ meaning that v₁ les v₂ for some v₁ isin S₁ and v₂ isin S₂. In this paper, we prove that an arbitrary reflexive relation can be thus represented. Similar representation theorems are proven for different versions of this relation.