Can preschoolers make pragmatic inferences based on the intonation of an utterance? Previous work has found that young children appear to ignore intonational meanings and come to understand contrastive intonation contours only after age six. We show that four-year-olds succeed in interpreting an English utterance, such as "It LOOKS like a zebra", to derive a conversational implicature, namely [but it isn't one], as long as they can access a semantically stronger alternative, in this case "It's a zebra". We propose that children arrive at the implicature by comparing such contextually provided alternatives. Contextually leveraged inferences generalize across speakers and contexts, and thus drive the acquisition of intonational meanings. Our findings show that four-year-olds and adults are able to bootstrap their interpretation of the contrast-marking intonation by taking into account alternative utterances produced in the same context.