The different systems of rough set theory treat uncertainty in special ways. There is a very important common property: all systems rely on given background knowledge and we cannot say more about an arbitrary set or about its members than its lower and upper approximations make possible. New membership relations have to be introduced. The semantics of classical logic is based on classical set theory. An important question: Is it possible to build logical systems with semantics relying on different versions of the theory of rough sets by using new membership relations? The paper gives an overview of first-order logical systems with partial semantics in order to show the influence of incomplete/uncertain information in logically valid inferences.