This paper is concerned with the use of voting procedures by groups of individuals to produce a social choice. Several new conditions that voting systems might be required to meet are suggested in this paper. These criteria are varying expressions of the notion that when there are more than two alternatives, social choice should be based on the results of majority voting on pairs. Five voting procedures or social choice functions are examined with respect to these conditions plus a condition requiring the social choice set to be externally stable. Two functions of the five examined are clearly better, according to these conditions, than the remaining three.