This paper considers a set of presupposition triggers including focus, questions, 'contrastive' statives, and an 'affirmation/negation' construction involving and not, where presuppositions are cancelable. It is proposed that these constructions, rather than having strict semantic presuppositions, have representations involving alternative sets in the sense of alternative semantics of questions and focus, and that a default process generates a presupposition from the alternative set. Presupposition projection facts are dealt with by stating a default constraint referring to dynamic denotations. The analysis can be extended to other constructions and lexical items with defeasible presupposition triggering behavior, such as inchoatives, by hypothesizing a representation involving alternatives.