We introduce and investigate a family of consequence relations with the goal of capturing certain important patterns of data-driven inference. The inspiring idea for our framework is the fact that data may reject, possibly to some degree, and possibly by mistake, any given scientific hypothesis. There is no general agreement in science about how to do this, which motivates putting forward a logical formulation of the problem. We do so by investigating distinct definitions of ‘rejection degrees’ each yielding a consequence (...) relation. Our investigation leads to novel variations on the theme of rational consequence relations, prominent among non-monotonic logics. (shrink)