The modifier almost indicates, intuitively, that the modified constituent is at least close to being true, but is not in fact true. For example, the sentence Bill almost swam the English Channel tells us (i) that Bill came close to swimming the Channel and (ii) that Bill did not swim the Channel. The status of the negative component in (ii) has been the subject of much debate since at least Sadock (1981), owing partly to its tendency to disappear in the presence of other operators, similar to scalar implicatures, while also resisting cancellation by following discourse, quite unlike implicatures. This paper will develop an analysis of almost which maintains both components in the truth-conditional meaning while explaining this puzzling behaviour. This will require two assumptions: (i) that almost covertly takes scope above a downward-entailing operator under which it is embedded at surface structure, and (ii) that a contradictory logical form created by said scope can be rescued by a covert, exhaustifying operator exh, as has been proposed for other propositional operators (Crnič 2013).