A Realistic (non-associative) Interval Logic and How Interval Computations Help in Proving Results about It

Experts' uncertainty about their statements Si is described by probabilities pi. The conclusion C of an expert system normally depends on several statements Si, so to estimate the reliability p(C), we must estimate the probability of Boolean combinations like S1 &S2. We cannot ask experts about all 2 n (> 10 10) such combinations, so we must estimate p(S1&S2) based on p1 = p(S1) and p2 = p(S2). One can use the interval p = max(p1 +p2 ?1; 0); min(p1; p2)] of possible values of p(S1&S2), but this often leads to p(C) = 0; 1]. A natural idea is to use a midpoint of p instead; this midpoint is a mathematical expectation of p(S1&S2) over a uniform (second order) distribution on all possible probability distributions. This midpoint operation & is not associative (which ts well with human reasoning). We show that some properties of this operation, like semi-associativity and the upper bound (1/9) on the diierence a&(b&c) ? (a&b)&c, can be derived by using interval computations.