Despite the ubiquity of conventional implicatures in language and the critical role they play in argumentation, they have heretofore been almost entirely absent from theories of argument and the linguistic expression of reasoning. In this paper, we discuss conventional implicatures (cis) as an interesting phenomenon at the interface of semantics, pragmatics and argumentation by harnessing research in semantics and pragmatics and extending an existing account for argument diagramming with this type of implicit meaning. In particular, we show that cis are unlike enthymemes, which are extremely challenging to conceptualise and to specify precisely. Instead, cis are anchored on the linguistic surface, trigger a largely predictable discourse contribution and are therefore more apt for argument analysis. By surfacing conventionally implicated material, we can unpack a wider variety of ways in which arguments are triggered by, composed of, and demolished by implicit discourse material, in particular inferential structures, conflicts and references to ethos. This also allows us to model the complex interplay between conventional implicature and argumentation, which in turn sheds new light on the interplay of meaning and argumentation in general.