 As it turns out, this problem for Bi-OT seems larger than it really is. Intuitively, an answer like (2) suggests that the speaker has incomplete information (she doesn’t know who of John or Bill came). But the interpretations that we considered so far are world states that do not encode different amounts of speaker knowledge. So, to take this into account in Bi-OT (or in any other analysis of Quantity implicatures) we should allow for alternative interpretations that represent different knowledge states of the speaker. Aloni (2007) gives a Bi-OT account of ignorance implicatures (inferences, like the above, that the speaker lacks certain bits of possibly relevant information), alongside indifference implicatures (that the speaker does not consider bits of information relevant enough to convey). Moreover, it can be shown that, as far as ignorance implicatures are concerned, the predictions of Bi-OT line up with the pragmatic interpretation function called ‘Grice’ in various (joint) papers of Schulz and Van Rooij (e.g., Schulz &amp; Van Rooij, 2006). In these papers it is claimed that Grice implements the Gricean maxim of Quality and the first maxim of Quantity, and it is shown that in terms of it (together with an additional assumption of competence) we can account for many conversational implicatures, including the ones of (1) and (2).