Scalar implicatures arise when a speaker uses a logically weak alternative in a context where a logically stronger alternative was available. Presuppositional implicatures, as I call them, arise when a speaker uses a presuppositionally weak alternative when a presuppositionally stronger alternative was available. My goal is to give a detailed, working theory of presuppositional implicatures, and show that they are a special case of scalar implicatures. In doing so, I carefully contrast presuppositional implicatures with antipresuppositions. These two phenomena have been treated as closely related in the literature, but some differences have not been adequately appreciated. Antipresuppositions are observed when a presuppositionally weak alternative is infelicitously used in a context that satisfies the presupposition of a presuppositionally stronger alternative. Presuppositional implicatures arise when a presuppositionally weak alternative is felicitously used in a context that does not satisfy the presupposition of a presuppositionally stronger alternative, but where that stronger presupposition would have been accommodated. Attention to this difference reveals a shortcoming in Schlenker’s (Nat Lang Semant 20:391–429, 2012) theory of presuppositional implicature. This paper both identifies and remedies that shortcoming.