The history of modality and modal logic saw many changes in the twentieth century. After a period of dismissal and disparage with Carnap’s quasi-possible-worlds, i.e. the linguistic state-descriptions, a new era were opened up. Things happened quickly: after Kripke introduced possible-worlds as models and/or indices, David Lewis developed his theory of concrete possible worlds as a full-blooded metaphysical theory. Many were opposing this move (as many do still today), but after a while, it turned out that merely possible worlds are not sufficient to account for our modal idioms and truths. Impossible worlds are needed as well. There are many options how to react to this move (from entire rejection to partial acceptance), but the aim of this presentation is to present a dilemma for impossible-world theorists. After discussing some usual puzzles and problems (some of which could be solved by the late Carnap’s meaning-postulate method), I will argue that if one were to accept contradictory sentences and their logical formulation, two options remain: either (1) hold that a certain world (or a states within it) makes them true and, in that case, one is committed to a strong logical realism; or (2) those sentences are true in virtue of their meanings and formation rules, and in that case, logic is detached from the extra-linguistic world. Given the problematic character of (1), (2) seems to be the viable option, but it does not require the existence of concrete impossible worlds.