The problem of aggregating individual preference orderings to form a social ordering took a new turn when Arrow organized the subject abstractly. We study here his celebrated theorem that five plausible conditions on the method of aggregation are inconsistent. This theorem is in fact false in general, as a counter-example shows. When we increase the amount of disagreement which is allowed to occur, then the inconsistency is restored. The modified result preserves much of the impact of the original theorem.