Uncertainty calculus with verbally defined belief‐intervals

The intended purpose of the present article is two‐fold: first, introducing an interval‐like representation of uncertainty that is an adequate summary of the following two items of information: a report on how strongly the validity of a proposition is supported by a body of evidence and a report on how strongly the validity of its negation is supported. A representation of this type is called a beliefinterval and is introduced as a subinterval of a certain verbal scale consisting of nine linguistic estimates expressing the amount of support provided for the validity of a proposition and/or its negation; each linguistic estimate is represented as a fuzzy number in the interval [0,1]. A belief‐interval is bounded from below by an estimate indicating the so‐called degree of support and from above by an estimate indicating the so‐called degree of plauibility. The latter is defined as the difference between a fuzzy number representing the maximal degree of support that might be provided for a proposition in general and a fuzzy number expressing the degree of support provided for the validity of the negation of the proposition under consideration. The so‐introduced degrees of support and plausibility of a proposition are subjective measurements provided by the expert on the basis of some negative and/or positive evidence available to him. Thus, these two notions do not have the same measure‐based origins as do the set‐theoretic measures of support and plausibility proposed by G. Shafer, neither do they coincide with the possibility and necessity measures proposed by L. Zadeh. The main difference is that in our case the degree of plausibility might be, in cases of contradictory beliefs, less than its corresponding degree of support. Three types of belief‐intervals are identified on the basis of the different amounts of support that might be provided for the validity of a proposition and/or its negation, namely balanced, unbalanced, and contradictory belief‐intervals. The second objective of this article is to propose a calculus for the belief‐intervals by extending the usual logical connectives and, or, negation, and implies. Thus, conjunctive and disjunctive operators are introduced using the Dubois' parametrized family of T‐norms and their dual T‐conorms. The parameter Q characterizing the latter is being interpreted as a measure of the strength of these connectives and further interpretation of the notion of strength is done in the cases of independent and dependent evidence. This leads to the introduction of specific conjunctive and disjunctive operators to be used separately in each of the latter two cases. A negation operator is proposed with the main purpose of determining the belief‐interval to be assigned to the negation of a particular proposition, given the belief‐interval of the proposition alone. A so‐called aggregation operator is introduced with the purpose of aggregating multiple belief‐intervals assigned to one and the same proposition into a total belief‐interval for this particular proposition. Detachment operators are proposed for determining the belief‐interval of a conclusion given the belief‐interval of the premise and the one that represents the amount of belief commited to the validity of the inference rule itself. Two different detachment operators are constructed for use in cases when: (1) the presence of the negation of the premise suggests the presence of the negation of the conclusion, and (2) when the presence of the negation of the premise does not tell anything at all with respect to the validity of the conclusion to be drawn.