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Suppose we are given a set of exact conjunctive views V and a conjunctive query Q. Suppose we wish to answer Q using V, but the classical test for the existence of a conjunctive rewriting of Q using V answers negatively. What can we conclude: (i) there is no way Q can be answered using V, or (ii) a more powerful rewriting language may be needed. This has been an open question, with conventional wisdom favoring (i). Surprisingly, we show that the right answer is actually (ii). That is, even if V provides enough information to answer Q, it may not be possible to rewrite Q in terms of V using just conjunctive queries – in fact, no monotonic language is sufficiently powerful. We also exhibit several well-behaved classes of conjunctive views and queries for which conjunctive rewritings remain sufficient. This continues a previous investigation of rewriting and its connection to semantic determinacy, for various query and view languages.