In the first part of this paper I consider the Gricean account of communication, as structured by the Cooperative Principle and the four maxims. Several authors, including Jean Goodwin [10], Fred Kauffeld [17], Michael Gilbert [7], Ernie Lepore and Mathew Stone [22], among others, argue that the Gricean view of communication fails in as much as it pretends to offer an account of all such human interactions. As Goodwin and Kauffeld suggest, a more promising starting point is to consider the variety of contextually determined presumptions that we make about speakers and that we rely upon in interpreting utterances. These presumptions are established in various ways, and are dropped, or defeated, in certain conditions. In order to clarify these aspects we need to inquiry into the nature of presumptions. I argue that Kauffeld’s [18], [19], [20] account of presumptions is useful in this context. In the second part of the paper I look at what this account tells us about how, and in what conditions, presumptions in communication are rebutted.