Conjunctive interpretation of disjunction

In this extended commentary I discuss the problem of how to account for "conjunctive" readings of some sentences with embedded dis-junctions for globalist analyses of conversational implicatures. Following Franke (2010, 2009), I suggest that earlier proposals failed, because they did not take into account the interactive reasoning of what else the speaker could have said, and how else the hearer could have interpreted the (alternative) sentence(s). I show how Franke's idea relates to more traditional pragmatic interpretation strategies.