A Counter Example to Theorems of Cox and Fine

TL;DR

This paper presents a counterexample demonstrating that Cox's theorem, which justifies probability theory, does not hold in finite domains and challenges assumptions underlying its proof.

Contribution

It provides the first known counterexample to Cox's theorem and shows that Cox's assumptions are insufficient even in infinite domains.

Findings

Cox's theorem fails in finite domains

Counterexample disproves Fine's result on comparative probability

Cox's assumptions are inadequate for deriving probability rules

Abstract

Cox's well-known theorem justifying the use of probability is shown not to hold in finite domains. The counterexample also suggests that Cox's assumptions are insufficient to prove the result even in infinite domains. The same counterexample is used to disprove a result of Fine on comparative conditional probability.