Some presuppositions are easier to cancel than others in embedded contexts. This contrast has been used as evidence for distinguishing two fundamentally different kinds of presuppositions, ‘soft’ and ‘hard’. ‘Soft’ presuppositions are usually assumed to arise in a pragmatic way, while ‘hard’ presuppositions are thought to be genuine semantic presuppositions. This paper argues against such a distinction and proposes to derive the difference in cancellation from inherent differences in how presupposition triggers interact with the context: their focus sensitivity, anaphoricity, and question–answer congruence properties. The paper also aims to derive the presuppositions of additive particles such as too, also, again, and of it-clefts.