Abstract
We propose two simple logics which attempt to draw sensible conclusions about the world on the basis of contradictory information, one of them defined semantically and the other defined syntactically. Their fundamental feature is that the premises are interpreted in the four-valued Belnap-Dunn logic, whereas the conclusions are interpreted classically.
These logics combine certain aspects of paraconsistent and non-monotonic logics, but reject some principles common to both. In particular, in this context it is not appropriate to assume reflexivity (i.e. to assume that p follows from p).