People often deny having meant what the audience understood. Such denials occur in both interpersonal and institutional contexts, such as in political discourse, the interpretation of laws and the perception of lies. In practice, denials have a wide range of possible effects on the audience, such as conversational repair, reinterpretation of the original utterance, moral judgements about the speaker, and rejection of the denial. When are these different reactions triggered? What factors make denials credible? There are surprisingly few experimental studies directly targeting such questions. Here, we present two pre-registered experiments focusing on (i) the speaker’s incentives to mislead their audience, and (ii) the impact of speaker denials on audiences’ moral and epistemic assessments of what has been said. We find that the extent to which speakers are judged responsible for the audience’s interpretations is modulated by their (the speakers’) incentives to mislead, but not by denials themselves. We also find that people are more willing than we expected to revise their interpretation of the speaker’s utterance when they learn that the ascribed meaning is false, regardless of whether the speaker is known to have had incentives to deceive their audience. In general, these findings are consistent with the idea that communicators are held responsible for the cognitive effects they trigger in their audience; rather than being responsible for, more narrowly, only the effects of what was ‘literally’ said. In light of our findings, we present a new, cognitive analysis of how audiences react to denials, drawing in particular on the Relevance Theory approach to communication. We distinguish in particular: (a) the spontaneous and intuitive re-interpretation of the original utterance in light of a denial; (b) the attribution of responsibility to the speaker for the cognitive effects of what is communicated; and (c) the reflective attribution of a particular intention to the speaker, which include argumentative considerations, higher-order deniability and reputational concerns. Existing experimental work, including our own, aims mostly at (a) and (b), and does not adequately control for (c). Deeper understanding of what can be credibly denied will be hindered unless and until this methodological problem is resolved.