A Note on Fine's Axioms for Qualitative Probability

Abstract

Fine gives axioms on a binary relation $\precsim$ on a field of events, with $A \precsim B$ interpreted as "$A$ is (subjectively) no more probable than $B$," sufficient to guarantee the existence of an order-preserving probability measure and an additive order-preserving probability measure. It is noted that one of Fine's axioms, that the order topology have a countable base, can be replaced by the more appealing axiom that there is a countable order-dense subset.