Summary: One of the most significant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work, we propose means for drawing conclusions from systems that are based on classical logic, although the information might be inconsistent. The idea is to detect those parts of the knowledge base that ‘cause’ the inconsistency, and isolate the parts that are ‘recoverable’. We do this by temporarily switching into Ginsberg/Fitting multivalued framework of bilattices (which is a common framework for logic programming and nonmonotonic reasoning). Our method is conservative in the sense that it considers the contradictory data as useless and regards all the remaining information unaffected. The resulting logic is nonmonotonic, paraconsistent, and a plausibility logic in the sense of Lehmann.