According to the Presupposition-Denial Account, complement set reference arises when focus is on the "shortfall" between the amount conveyed by a natural language quantifier and a larger, expected amount. Negative quantifiers imply a shortfall, through the denial of a presupposition, whereas positive quantifiers do not. An exception may be provided by irony. One function of irony is to highlight, through indirect negation, the shortfall between what is expected/desired, and what is observed. Thus, a positive quantifier used ironically should also lead to a shortfall and license complement set reference. Using ERPs, we examined whether reference to the complement set is more felicitous following a positive quantifier used ironically than one used non-ironically. ERPs during reading showed a smaller N400 for complement set reference following an ironic compared to a non-ironic context. The shortfall generated thorough irony is sufficient to allow focus on the complement set, supporting the Presupposition-Denial Account.