In this paper we present a general logical framework for (weighted) MAX-SAT problem, and study properties of inference rules for branch and bound MAX-SAT solver. Several rules, which are not equivalent but ?-equivalent, are proposed, and we show that ?-equivalent rules are also sound. As an example, we show how to exploit inference rules to achieve a new lower bound function for a MAX-2-SAT solver. Our new function is admissible and consistently better than the well-known lower bound function. Based on the study of inference rules, we implement an efficient solver and the experimental results demonstrate that our solver outperforms the most efficient solver that has been implemented very recently [Heras and Larrosa, 2006], especially for large instances.