Arrow already knew that U is a stronger domain condition than is needed for an impossibility result. The free triple property and the chain property are weaker conditions that replace U in some versions of Arrow’s theorem (Campbell and Kelly 2002). These versions, being more informative, are, from a logical point of view, better. U is simpler to state than their domain conditions, though, and might be found more intuitive. Notice that the weaker domain conditions still require a lot of variety among profiles. A typical proof of an Arrow-style impossibility theorem requires that the domain is unrestricted with respect to some three alternatives. In this case there is always a preference profile like the one implicated in the paradox of voting in Section 1, from which pairwise majority decision derives a cycle.