The standard way of answering queries over incomplete databases is to compute certain answers, defined as the intersection of query answers on all complete databases that the incomplete database represents. But is this universally accepted definition correct? We argue that this "one-size-fits-all" definition can often lead to counterintuitive or just plain wrong results, and propose an alternative framework for defining certain answers. The idea of the framework is to move away from the standard, in the database literature, assumption that query results be given in the form of a database object, and to allow instead two alternative representations of answers: as objects defining all other answers, or as knowledge we can deduce with certainty about all such answers. We show that the latter is often easier to achieve than the former, that in general certain answers need not be defined as intersection, and may well contain missing values in them. We also show that with a proper choice of semantics, we can often reduce computing certain answers - as either objects or knowledge - to standard query evaluation. We describe the framework in the most general way, applicable to a variety of data models, and test it on three concrete relational semantics of incompleteness: open, closed, and weak closed world. (C) 2015 Elsevier B.V. All rights reserved.