Avoiding foregone conclusions : geometric and foundational analysis of paradoxes of finite additivity.

Abstract

We consider an example in which the use of finitely additive probabilities leads to the seemingly paradoxical conclusion that we may reason to a foregone conclusion. We give an informal treatment of this paradox. We then develop a geometric approach which formalises this treatment, and offers a foundational resolution of the problem. We discuss this approach both in the context of assessing the value of such paradoxical data and also when used for finite approximation.