 Objection 1: There is no motivation for the appeal to Euclidean distance. Leitgeb and Pettigrew show that the only inaccuracy measure that satisfies  Global Normality and Dominance,  together with their other conditions on inaccuracy measures, is the Brier score, which we defined above. That is, imposing these conditions entails  Brier Inaccuracy.  The problem with this characterization, however, is that it depends crucially on the appeal to the Euclidean distance made in  Global Normality and Dominance,  and no reason is given for appealing to the Euclidean distance measure in particular, rather than some other measure of distance between credence functions. Suppose we replace that condition with one that says that a legitimate global inaccuracy measure must be a strictly increasing function of the so-called Manhattan or city block distance measure, where the distance between two credence functions measured in this way is defined as follows:  \[||c - c'||_1 := \sum_{X \in \mathcal{F}} |c(X) - c'(X)|\] That is,   the Manhattan distance between two credence functions is obtained by summing the differences between the credences they each assign to the various propositions on which they are defined. Together with the other constraints that Leitgeb and Pettigrew place on inaccuracy measures, this alternative constraint entails that the only legitimate inaccuracy measure is the so-called absolute value score, which is defined as follows:   \[\mathfrak{A}(c, w) := \sum_{X \in \mathcal{F}} |v_w(X) - c(X)|\]