 Incommensurability is most directly a challenge to Completeness, since on the most natural interpretation of \(\succcurlyeq,\) the fact that \(A\) and \(B\) are incommensurable means that neither \(A \succcurlyeq B\) nor \(B \succcurlyeq A.\) But incommensurability can instead be framed as a challenge to Transitivity, if we assume that incommensurability is indifference, or define \(A \succcurlyeq B\) as the negation of \(B \succ A\) (thus assuming Completeness by definition). To see this, notice that if two options \(A\) and \(B\) are incommensurable, then “sweetening” \(A\) to a slightly better \(A^+\) will still leave \(A^+\) and \(B\) incommensurable. For example, if \(A\) is a career in the military and \(A^+\) is this career but with a slightly higher salary, the latter is still incommensurable with a career as a priest. This pattern suffices to show that the relation \(\sim\) is intransitive, since \(A \sim B\) and \(B \sim {A^+},\) but \({A^+} \succ A\) (de Sousa 1974).