A disjunctive sentence like Olivia took Logic or Algebra suggests that Olivia didn’t take both classes (‘exclusivity’ inference) and that the speaker is ignorant as to which of the classes she took (‘ignorance’ inference). The corresponding sentence with a possibility modal, Olivia can take Logic or Algebra, conveys that she can take Logic and that she can take Algebra (‘free choice’ inference). These exclusivity, ignorance and free choice inferences are argued by many to be scalar implicatures. Recent work has looked at cases in which exclusivity and ignorance appear to be computed in the presupposition, in addition to or independently from the assertion. On the basis of those data, Spector & Sudo (2017) has argued for a hybrid account based on two distinct scalar strengthening mechanisms, namely a grammatical theory of implicatures for deriving implicatures in the assertion and a pragmatic principle for deriving implicatures in the presupposition. In this paper, we observe that a sentence like Noah is unaware that Olivia can take Logic or Algebra has a reading on which free choice appears in the presupposition (i.e., it suggests that Olivia can choose between the two classes), but not in the assertion (i.e., it conveys that Noah doesn’t believe that she can take either one). We show that deriving this presupposed free choice reading is challenging on Spector & Sudo’s (2017) hybrid account. Following the dialectic in Fox 2007, we argue against a pragmatic approach to presupposition-based implicatures on the ground that it is not able to account for presupposed free choice. In addition, we raise a novel challenge for Spector & Sudo’s (2017) system coming from the conflicting presupposed ignorance triggered by sentences like #Noah is unaware that I have a son or a daughter, which is odd even if it’s not common knowledge whether the speaker has a son or a daughter. More generally, our set of data reveals systematic parallelism between the assertion and presuppositional levels in terms of exclusivity, ignorance, and free choice. We argue that such parallels call for a unified analysis of those inferences at the two different levels at which they can arise, and we sketch how a grammatical theory of implicatures where meaning strengthening operates in the same way at both levels (Gajewski & Sharvit 2012, Magri 2009 and Marty 2017) can account for such parallels.