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Two mouse-tracking experiments tested predictions from two different models of scalar implicature as to whether exhaustive interpretations are computed prior to ignorance implicatures. We use different German intonational patterns to probe the availability of these interpretations (Experiments 1 and 2) and add a speaker competence manipulation in Experiment 2. Results from Experiment 1 found that deriving exhaustive interpretations with an L+H* was delayed to ignorance implicatures with an L*+H contour. Experiment 2 replicated this finding even with a strengthened competence assumption about the speaker. We interpret our processing data as providing constraints on the computational mechanisms underlying the interpretation of scalar implicatures.