 To verify that median grading satisfies the reformulated conditions, take for instance the nondictatorship condition. A person \(d\) is a dictator of \(f\) if for any alternatives \(x\) and \(y\), and for any profile \(\langle\ldots, G_{d},\ldots\rangle\) in the domain of \(f\): if \(xP_{d}y\), then \(xPy\). When \(f\) is median grading, this means that whenever a dictator expresses a strict preference for one alternative over another, by giving it a higher grade, this alternative also ranks strictly higher in the social ordering, having a higher median grade. Except in the trivial case of a language with only a single grade (which makes it impossible to express strict preferences, giving everybody vacuous dictatorial powers), and provided there are at least two alternatives and three people, median grading on an unrestricted domain has no dictator. To see that it is so, consider any given person \(p\), and any two alternatives \(A\) and \(B\). In an unrestricted domain there is some profile in which \(p\)’s grade for \(A\) is higher than \(p\)’s grade for \(B\), but in which everybody else’s grades for these alternatives are just the other way around: everybody other than \(p\) gives to \(B\) the same grade that \(p\) gives to \(A\), and gives to \(A\) the same grade that \(p\) gives to \(B\). In this profile, the median grade for \(A\) is \(p\)’s grade for \(B\), and the median grade for \(B\) is \(p\)’s grade for \(A\), which is higher. So although \(p\) strictly prefers \(A\) to \(B\), \(B\) ranks strictly higher than \(A\) in the social ordering, and \(p\) is not a dictator. Repeating this demonstration for each person, no one else is a dictator either. Satisfaction of the other reformulated conditions on the unrestricted domain of grade profiles is similarly straightforward.