In this paper we present our implementation of a system of active logic that processes natural language discourses. We focus on problems that involve presupposition and the associated well-known problems of the projection of presupposition. We discuss Heim's largely successful theory of presupposition and point out certain limitations. We then use these observations to build our discourse processor based on active logic. Our main contributions are the handling of problems that go beyond the scope of Heim's theory , especially discourses the involve cancellation of presupposition. Ongoing work suggests that conversational implicature and the cancellation of implicature can also be treated by our methods. Key words: presupposition, discourse, con text, accommodation, active logic, implicature. (Also cross-referenced as UMIACS-TR-96-62)