Presupposition is one way of expressing irony which is a communicative behavior in that the literal meaning and background assumptions associated with an utterance differ in systematic ways from the reality as assumed by the interlocutors. Typically, the intended meaning of ironical utterances is the opposite of the literal meaning. Irony is perceived through a complex interaction between an utterance and its context and sense many social functions. The purpose of this paper is to explore the presupposition role in the recognition of irony in Middleton (1989) cynical tragedy. The issue of controversy in irony research depends on what features of irony people endorse. Thus, a more comprehensive view of irony, the implicit display theory of verbal irony is herewith proposed. Discussion of the findings suggests that implicit display theory and the tool of presupposition can account for indirect effects on the degree of irony.