The concept of truth is at the core of science, journalism, law, and many other pillars of modern society. Yet, given the imprecision of natural language, deciding what information should count as true is no easy task, even with access to the ground truth. How do people decide whether a given claim of fact qualifies as true or false? Across two studies (N = 1181; 16,248 observations), participants saw claims of fact alongside the ground truth about those claims. Participants classified each claim as true or false. Although participants knew precisely how accurate the claims were, participants classified claims as false more often when they judged the information source to be intending to deceive (versus inform) their audience, and classified claims as true more often when they judged the information source to be intending to provide an approximate (versus precise) account. These results suggest that, even if people have access to the same set of facts, they might disagree about the truth of claims if they attribute discrepant intentions to information sources. Such findings may shed light on the robust and persistent disagreements over claims of fact that have arisen in the “post-truth era”.