Foundations for conditional probability

Abstract

The main result presented in this article is that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities. This is justified by the fact that probability is coherent as confirmed by its common formalizations, and by our result that a function is coherent if and only if it is a subset of conditional expectation induced by a plausible preorder on random quantities. In addition to offering a different perspective on conditional probability, our use of a plausible preorder in the role of a fundamental notion extends conditional probability to cases in which the calculation of conditional probability using the P(A|C)=\frac{P(A\wedge C)}{P(C)} rule fails: if P is a coherent function, then it can be extended so that for every event A and nonzero event C holds that P(A|C)=0 if A\wedge C=0 and P(A|C)=1 if A\wedge C=C, no matter whether the unconditional probability P(C) is zero or whether it is defined.