Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion, and since in practice consistency is dificult to achieve there are many potential applications of paraconsistent logics in query answering systems. We compare the paraconsistent and the non-monotonic solutions to the problem of contradictions. We propose a many-valued paraconsistent logic based on a simple notion of indeterminacy. In particular we describe the semantics of the logic using key equalities for the logical operators. We relate our approach to works on bilattices. We also discuss and provide formalizations of two case studies, notably the well-known example involving penguins and a more interesting example in the domain of medicine.