Theoretical and empirical research on quantity implicature has concurred that pragmatically strengthened, richer readings are not available when they are not relevant to the discourse purpose. However, this claim relies on an appeal to a notion of "relevance" which has proved difficult to make precise. In this paper I discuss and contrast two potential contributory factors to relevance: adherence to the QUD, and form-based priming effects. The former can be considered to operate at a relatively high level of analysis, from the speaker's perspective, and influences the semantic content that the speaker should be attempting to convey, while the latter is assumed to reflect low-level psychological preferences and influences the form of words that the speaker should use. I argue that pragmatics, and specifically implicature, constitutes a useful testbed for distinguishing these effects – the availability of an implicature can be used as an indicator that a particular stronger alternative would also have been an acceptable utterance, while its unavailability suggests that the stronger alternative would not necessarily have been acceptable. I discuss recent experimental data from this perspective, and argue that both QUD and priming effects are customarily at play. I conclude by exploring the implications of this for our view of pragmatics and its interfaces.