 Given various assumptions about the richness of the preference space, and certain ‘consistency assumptions’, he can define a real-valued utility function of the outcomes \(A, B\), etc — in fact, various such functions will represent the agent’s preferences. He is then able to define equality of differences in utility for any outcomes over which the agent has preferences. It turns out that ratios of utility-differences are invariant — the same whichever representative utility function we choose. This fact allows Ramsey to define degrees of belief as ratios of such differences. For example, suppose the agent is indifferent between \(A\), and the gamble “\(B\) if \(X, C\) otherwise”. Then it follows from considerations of expected utility that her degree of belief in \(X, P(X)\), is given by: