The notion of a bilattice was rst proposed by Ginsberg as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting. In the present paper we develop proof systems, which correspond to bilattices in an essential way. We then show how to use those bilattices for efcient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii concerning inconsistencies, which happen to suit well the framework of bilattices. The outcome is a paraconsistent logic with a lot of desirable properties.