I explain what exactly constrains presupposition projection in compound sentences and argue that the presuppositions that do not project are conditionalized, giving rise to inferable conditional presuppositions. I combine elements of and which, together with an additional, independently motivated assumption, make it possible to construct an analysis that makes correct predictions. The core of my proposal is as follows: When a speaker felicitously utters a compound sentence whose constituent clauses require presuppositions, the hearer will infer that the speaker presupposes those (...) propositions, unless the sentence contains some element that makes the hearer realize that, if the speaker actually presupposed them, she would be either uninformative or inconsistent in her beliefs. In these cases, the propositions that would have been presupposed, had the clauses been uttered in isolation, will not be presupposed, i.e. the clausal presuppositions will not project. (shrink)