Bearing contradictory knowledge is often unavoidable among multi-agents. Measuring inconsistency degrees of knowledge bases of different agents facilitates the understanding of an agent to her environment. Several semantics or syntax-based approaches have been proposed to quantify inconsistencies. In this paper, we propose a new inconsistency measuring framework based on both minimal unsatisfiable sets and maximal consistent sets. Firstly, we define a graph representation of knowledge bases, based on which we furthermore explore the logical property of the Additivity condition. Then, we show how the structure of the proposed graph representation can be used to discriminate, in a fine-grained way, the responsibility of each formula or a set of formulae for the inconsistency of a knowledge base. Finally, we extend our framework to provide an inconsistency measure for a whole knowledge base. All the proposed measures are shown satisfying the desired properties.