 Figure 1: In this figure, we plot the various possible credence functions defined on a proposition Heads and its negation Tails in the unit square. Thus, we plot the credence in Heads along the horizontal axis and the credence in Tails up the vertical axis. We also plot the vindicated credence functions \(v_{w_1}\) and \(v_{w_2}\) for the two worlds \(w_1\) (at which Tails is true and Heads is false) and \(w_2\) (at which Heads is true and Tails is false). The diagonal line between them contains all and only the credence functions on these two propositions that are probability functions and thus satisfy Probabilism. \(c^*\) (which assigns 0.7 to Heads and 0.6 to Tails) violates Probabilism. The lower right-hand arc contains all the credence functions that are exactly as inaccurate as \(c\) at world \(w_2\), where that inaccuracy is measured using the Brier score. To see this, note that the Brier score of \(c^*\) at \(w_2\) is the square of the Euclidean distance of the point \(c^*\) from the point \(v_{w_2}\). Thus, the credence functions that have exactly the same Brier score as \(c^*\) at \(w_2\) are those that lie equally far from \(v_{w_2}\). For the same reason, the upper left-hand arc contains all the credence functions that are exactly as inaccuracy as \(c\) at world \(w_1\). Every credence function that lies between the two arcs is more accurate than \(c^*\) at both worlds. These are the ones whose squared Euclidean distance from \(v_{w_2}\) is less than the squared Euclidean distance of \(c^*\) from \(v_{w_2}\), and similarly for \(v_{w_1}\). It assigns 0.55 to Heads and 0.45 to Tails. \(c'\) is such a credence function. \(c'\) also satisfies Probabilism. 