The interior of inconsistency in a knowledge base

Looking inside local inconsistencies arising in different parts of a knowledge base may help us better frame the inconsistency of the knowledge base. Moreover, as possible changes of the inconsistency due to removing some formulas from the knowledge base, local inconsistencies play an important role in identifying contributions of formulas to the inconsistency in the knowledge base. In this paper, we focus on local inconsistencies arising in all inconsistent subsets of a knowledge base and their relations with the inconsistency in the whole knowledge base. We call inconsistencies in all the inconsistent subsets of a knowledge base the interior of inconsistency of the knowledge base, and characterize the interior of inconsistency in two directions. One is the distribution of local inconsistencies over the power set of the knowledge base and relations between them. It focuses on local appearances of the inconsistency in different parts of the knowledge base. The other is the hierarchical structure of the interior of inconsistency due to different deviations of local inconsistencies from the inconsistency of the knowledge base. This direction is more interested in the aspect of each local inconsistency as a potential change of the inconsistency due to removing some formulas. Then we consider the interior of inconsistency of a knowledge base from syntactic and semantic perspectives, respectively.