Measure Representations for Events in a Partial Order : an Axiomatization

Currently all the axiom systems for probability representation include the assumption that the ordering of events is total. However, empirical systems exist for which this condition is too strong. In particular, there may exist events A l , A2 in an algebra A for which the ordering is not connected; i.e., it is not possible to judge with complete confidence whether Al $A A2 or One approach to deriving representations for such sets proceed as follows: assume the existence of a uniformly distributed random variable which induces a a-field C containing ordered events C., i ~ I. For each A E A, half-open 1. intervals are formed, S .. = (C. < A < C.) for all i, j such that i < j, where 1.J 1 0 -0 J ~o is an ordering between the event A and the events C., C. E C. Consider for 1 J each A E A the triple (S, ~S' D), where S is a collection of S.. , for all i and 1J j, ~S is a total order to be read as "not more confident that" and 0 is a concatenation operation. Plausible axioms about the triple lead to a distribution of probabilities lPA for each A E A. MEASURE REPRESENTATIONS FOR EVENTS IN A PARTIAL ORDER: AN AXIOMATIZATION ROGER A. BLAU1 , RICHARD H. SHACHTMAN2 , THOMAS S. WALLSTEN 3