© 2018 Institute for Cognitive Science, Seoul National University. In a series of three experiments, we investigate ignorance inferences triggered by two numeral modifiers in Japanese, contrastive wa and sukunakutomo 'at least'. Experiment 1 (self-paced reading followed by acceptability judgments) investigates how Questions under Discussion (QuDs) affect ignorance inferences and shows that wa is QuD sensitive, while sukunakutomo is not. Given the results, we hypothesize that there is a competition between the two ways of expressing ignorance: sukunakutomo unambiguously encodes ignorance and is therefore preferred when the speaker wants to express uncertainty. In contrast, wa simply requires the presence of alternatives to the at-issue content, which are possible as far as the speaker is concerned. These alternatives can be about numbers, which trigger ignorance inferences, but need not be. Experiments 2 and 3 are forced binary choice tasks. Experiment 2 investigated how sensitive people are to ignorance of the speaker or contextual contrast when they are given a choice between wa and ga. Experiment 3 compared wa and sukunakutomo and supported the competition hypothesis. Specifically, Experiment 3 shows that sukunakutomo is preferred when the speaker is ignorant, while wa is preferred when the speaker is knowledgeable, and there is a contextually salient contrast that wa picks up on.