This paper takes up the problem from Stalnaker (1974) of deriving the pragmatic presuppositions of verbs such as know, stop and win as conversational implicatures, without hypothesizing a semantic presupposition. I interpret data discussed by Karttunen (1969), Chierchia and McConnell-Ginet (1990), Simons (2001) and others as indicating that there is a distinct group of ?soft? presupposition triggers whose pragmatic presuppositions, though systematic, are also contextdependent and easily suspendable. These are distinguished from ?hard? presuppositions triggers like it-clefts and too which on the assumptions of this paper introduce semantic presuppositions. These distinctions are defended in sections 1 and 2. Sections 3 and 4 review and criticize proposals from Stalnaker (1974) and Simons (2001) for deriving the pragmatic presuppositions of soft triggers as conversational implicatures. Section 5 introduces the hypothesis that the pragmatic presuppositions of soft triggers come from alternatives to lexical meanings, such as the alternative lose to win. A pragmatic presupposition is derived as the default assumption that some alternative is true. In section 6, the default existential presupposition of intonational focus is attributed to the same process. Section 7 proposes a systematic pragmatic derivation of a conversational implicature, using a specific default axiom called G, and a general pragmatic process of enrichment reasoning. Sections 8, 9, and 10 address the projection problem for the pragmatic presuppositions of soft triggers. It is shown that projection data for these triggers is the same as what is seen for hard triggers, which would seem to favor an analysis using semantic presuppositions. The puzzle is resolved by replacing G with a default generalization L which refers to the local information states manipulated by compositional semantics in dynamic compositional theories. Section 9 also considers general issues of the interface between pragmatics and compositional semantics. Section 11 shows that the derivation using L also deals with projection data for focus.