The ability to handle exceptions, to perform iterated belief revision and to integrate information from multiple sources are essential skills for an intelligent agent. These important skills are related in the sense that they all rely on resolving inconsistent information. We develop a novel and useful strategy for conflict resolution, and compare and contrast it with existing strategies. Ideally the process of conflict resolution should conform with the principle of Minimal Change and should result in the minimal loss of information. Our approach to minimizing the loss of information is to weaken information involved in conflicts rather than completely removing it. We implemented and tested the relative performance of our new strategy in three different ways. We show that it retains more information than the existing Maxi-Adjustment strategy at no extra computational cost. Surprisingly, we are able to demonstrate that it provides a computationally effective compilation of the lexicog...