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A disjunctive sentence like  Olivia took Logic or Algebra  conveys that Olivia didn’t take both classes ( exclusivity ) and that the speaker doesn’t know which of the two classes she took ( ignorance ). The corresponding sentence with a possibility modal,  Olivia can take Logic or Algebra , conveys instead that she can take Logic and that she can take Algebra ( free choice ). 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 instead at the presupposition level, independently from the assertion. On the basis of those data, Spector and Sudo (Linguist Philos 40(5):473–517, 2017) have argued for a hybrid account relying on 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, but not in the assertion, and we show that deriving this reading is challenging on Spector and Sudo’s (2017) hybrid account. Following the dialectic in Fox (Presupposition and implicature in compositional semantics, Palgrave, London, pp 71–120, 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 and Sudo’s (2017) account coming from the conflicting presupposed  ignorance  triggered by sentences like  #Noah is unaware that I have a son or a daughter , which is infelicitous even if it’s not common knowledge whether the speaker has a son or a daughter. More generally, our data reveals a systematic parallelism between the assertion and presupposition levels in terms of  exclusivity ,  ignorance , and  free choice . We argue that such parallels call for a unified analysis and we sketch how a grammatical theory of implicatures where meaning strengthening operates in a similar way at both levels (Gajewski and Sharvit in Nat Lang Semant 20(1):31–57, 2012; Magri in A theory of individual-level predicates based on blind mandatory scalar implicatures, MIT dissertation, 2009; Marty in Implicatures in the DP domain, MIT dissertation, 2017) can account for such parallels.

Presupposed free choice and the theory of scalar implicatures