The presuppositions inherited from the consequent of a conditional or the second disjunct of a disjunction oscillate between a conditional and a non-conditional inference, depending on the context. This is problematic for most theories of presupposition projection in the literature, which only predict a conditional presupposition for such sentences (among others). The general response to this problem, the so-called ‘Proviso Problem’ (Geurts 1996), is to assume that in addition to the basic conditional presupposition, a non-conditional inference can arise depending on the relationship between the antecedent/first disjunct and the presupposition of the consequent/second disjunct. We discuss data for which this solution makes the wrong predictions. Similar data have been taken by van der Sandt (1992), Geurts (1996) and Garcia-Odon (2012) to motivate the DRT-approach to presuppositions. Schlenker (2011), however, has raised various arguments against such an approach. We propose an alternative analysis, which doesn’t have those problems. In our analysis, the differing presuppositions are the result of a systematic ambiguity involving exhaustification in a trivalent semantics: a non-conditional presupposition obtains with exhaustification, and a conditional one without. Independently motivated plausibility considerations decide which reading is chosen with no direct selection of presuppositions needed. We discuss how this approach deals with the various cases of proviso and the predictions it makes for biconditional sentences.