 Default logic fails many of the tests for a logical relation that were introduced in the previous section. It satisfied Cut and Full Absorption, but it fails Cautious Monotony (and thus fails to be cumulative). In addition, it fails Distribution, a serious limitation that rules out reasoning by cases. For example, if one knows that Smith is either Amish or Quaker, and both Quakers and Amish are normally pacifists, one cannot infer that Smith is a pacifist. Default logic also fails to represent Pollock’s undercutting defeaters. Finally, default logic does not incorporate any form of the principle of Specificity, the principle that defaults with more specific prerequisites ought, in cases of conflict, to take priority over defaults with less specific prerequisites. Recently, John Horty (Horty 2007a, 2007b) has examined the implications of adding priorities among defaults (in the form of a partial ordering), which would permit the recognition of specificity and other grounds for preferring one default to another. In addition, Horty allows for defeasible reasoning about these priorities (the relative weights of various defaults) by means of higher-order default rules. Such defeasible reasoning about relative weights enables Horty to give an account of Pollock’s  undercutting defeaters:  an undercutting defeater is a triggered default rule that lowers the weight of the undercut rule below some threshold, with the result that the undercut rule can no longer be triggered.