If incommensurability is treated as indifference, or if 'A is at least as good as B' is defined as the negation of 'B is strictly better than A', then Completeness holds by definition
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, notic