This paper focuses on sentences like Nixon is guilty, if Haldeman is guilty too, first discussed by Soames (1982) and Karttunen and Peters (1979), which raise three problems. First, they are felicitous and do not appear to have presuppositions. However all major theories of presup- positions predict that they should presuppose what the antecedent presupposes (e.g., the sen- tence above should presuppose that Nixon is guilty). Second, there is a contrast between these sentences and the corresponding sentence-initial conditionals like if Haldeman is guilty too, Nixon is guilty. Finally, a way to solve the problem would be to locally accommodate the pre- supposition in the antecedent. However, this wrongly predicts tautological truth-conditions. In the case above, the predicted meaning could be paraphrased as “Nixon is guilty, if both Haldeman and Nixon are guilty.” As a solution to these three problems, I propose that the presupposition is nonetheless locally accommodated in the antecedent and furthermore that the sentence is also interpreted exhaustively, which gives rise to a non-presuppositional and non-tautological meaning analogous to Nixon is guilty, only if both Haldeman and Nixon are guilty. Furthermore, I argue that the degraded status of the sentence-initial case is an indepen- dent fact rooted in the topic-focus structure of sentence-final conditionals. Finally, the present proposal can also be extended to treat related non-presuppositional cases like I will go, if we go together.