The presuppositions of embedded constituents tend to become presuppositions of the matrix sentence in which they are contained. Different approaches to presupposition projection provide different resources for describing and explaining this (and related) generalizations. An important architectural distinction that divides frameworks is whether the projection mechanism can filter presuppositions such that embedded presuppositions appear in modified form at the root. Theories that can filter a presupposition need a mechanism for “unfiltering” it to account for the generalization that embedded presuppositions tend to be inherited at the root, even when the projection component delivers a filtered presupposition to the root. Theories that cannot filter a presupposition can account for the generalization but not for the observation that presuppositions can be filtered. We propose a synthesis that generally allows presuppositions to be filtered, and that connects unfiltering with accommodation mechanisms that keep track of the presuppositions of sentences bounded by the matrix sentence and embedded triggering sentence.