Characterization of the Almost Agreeing Probabilities for the Kraft-Pratt-Seidenberg Structures

Petros Hadjicostas


When ``'' we say that ``event is at least as probable as event ''. This is a qualitative probability judgment. A qualitative probability structure, , where is an algebra on the set and is a binary relation on , satisfies connectedness, transitivity, nontriviality, nonnegativity, and additivity. In 1959, Kraft, Pratt, and Seidenberg answered in the negative the question of whether every finite structure of qualitative probability has a finitely additive, order-preserving probability representation. They achieved that by constructing an ordering over a 32-element algebra, the powerset of a 5-atom set, that does not have a probability representation. Their ordering satisfies four crucial conditions that prevent the existence of strictly agreeing quantitative probability representations. In addition, the three authors showed that there is an almost agreeing probability for their example. They went even further and constructed an example that does not even have an almost agreeing probability representation. The purpose of this paper is to review the proofs of the three authors, and to characterize all almost agreeing probabilities for all qualitative probability structures on the powerset of a 5-atom set that satisfy the four crucial conditions I mentioned above. In doing so, I prove some theorems that facilitate showing that (with finite) satisfies additivity.