The problem is that that criterion combined with the other conditions becomes equivalent to there being no 1-man quorums. As the rest of the proof shows the neg

The problem is that that criterion combined with the other conditions becomes equivalent to there being no 1-man quorums. As the rest of the proof shows the negation of this implies that there is a dictator, who completely determines the result for all votes. I guess in a ballot system it might be pretty tricky to figure out who this person is, but in the end it just boils down to the order in which the votes are put into the sorting system. No while sortition has it's advantages it goes against most people's intuition of what a fair voting system is. And technically arrow's definition of voting systems is deterministic, although you could always shuffle the votes beforehand.