Axiomatizing the Logic of Decision

Abstract

Ethan Bolker’s theorem [1,2] on the representation of functions resembling quotients of measures underlies the logic of decision [5] in the same way in which Holder-like theorems on the repsentation of Archimedean ordered groups, semigroups, etc., are seen in [6] as underlying various foundational systems of measurement. In [3] and in Chapter 9 of [1] Bolker applies his theorem to a system akin to that of [5], proving existence and uniqueness of probability and utility functions for preference rankings that satisfy certain axioms. Here I want to clarify the framework of [5] and bring Bolker’s theorem to bear directly upon it. It should be noted that Bolker’s theorem predated [5] and made it possible.