 In  section 5.4.2,  we introduced Undominated Dominance and we stated Theorem 5, which says that every non-probabilistic credence function \(c^*\) defined on \(\mathcal{F}\) is accuracy dominated by a credence function \(c'\) defined on \(\mathcal{F}\) that is itself not accuracy dominated by a credence function \(c\) defined on \(\mathcal{F}\). But you might think that this is still not sufficient to establish Probabilism. After all, while \(c'\) is not itself dominated by a credence function defined on \(\mathcal{F}\), it might be accuracy dominated by a credence function \(c^\dag\) defined on some other set of propositions \(\mathcal{F}^\dag\). For instance, take the non-probabilistic credence function \(c^*\) defined on \(\mathcal{F} = \{X, \neg X\}\), where \(c^*(X) = 0.6 = c^*(\neg X)\). It is Brier dominated by \(c'(X) = 0.5 = c'(\neg X)\), since \(c^*\) has Brier score \(|1-0.6|^2 + |0-0.6|^2 = 0.52\) at both worlds in \(\mathcal{W_F}\) while \(c'\) has \(|1-0.5|^2 + |0-0.5|^2 = 0.5\) at both worlds. But \(c'\) is Brier dominated by \(c^\dag\) defined on \(\mathcal{F}^\dag = \{X\}\), where \(c^\dag(X) = 0.5\), since this has Brier score \(|1-0.5|^2 = |0-0.5|^2 = 0.25\) at both worlds. A natural reaction to this is to define the epistemic disutility function to be the average Brier score rather than the total Brier score: