This chapter presents the most influential linguistic approaches to presupposition. Going beyond the traditional analyses of the problem of presupposition projection, it also considers recent developments in linguistics that link the analysis of presuppositions to general processes of cognition and reasoning, such as attention, probabilistic reasoning, theory of mind, information structure, attitudes and perspectival structure. I discuss some outstanding questions: whether presuppositions form one coherent group or should be thought of as different types of phenomena, why we have presuppositions at all, and why we see the presuppositions that we see (aka the triggering problem). Overall, the chapter stresses the need to consider the intricacies of the interaction of presuppositions with the broader discourse context.

This chapter focuses on the role that discourse relations and structure play in a variety of phenomena of interest to semanticists and philosophers. Not only do discourse relations add semantic content above and beyond the individual propositions expressed by the utterances in a discourse, but they, and the complex structures to which they give rise, can influence the interpretations of individual utterances, having an effect on the very propositions the utterances are understood to express. In this chapter, we look in detail at how theories of discourse structure can be brought to bear on at-issue and non-at-issue content, using appositive relative clauses and discourse parenthetical reports as illustrations. We also discuss recent efforts to use discourse structure to model conversational goals and capture the subjective nature of discourse interpretation as well as recent work extending theories of discourse structure to multimodal discourse. Along the way, we emphasize the importance of corpus work in studying discursive phenomena and raise a series of large questions to be pursued in future work.