Managing semantic heterogeneity is a complex task. One solution involves matching like terms to each other. We view Match as an operator that takes two graph-like structures (e.g., concept hierarchies or ontologies) and returns a mapping between the nodes of the graphs that correspond semantically to each other. State of the art matching systems (e.g., COMA, Cupid) perform well for many real world applications. However, matching systems may produce mappings that may not be intuitively obvious to human users. Moreover, there are cases where matching systems do not produce a useful mapping. In order for users to trust the mappings (and thus use them), they need to understand them. Also, if a system does not provide a mapping or provides a partial mapping, users need to understand answers so that they can understand either why a mapping was not produced or why only a partial answer was produced. In this paper we describe how matching systems can explain their answers using the Inference ...