Two alternatives for disjunction: An inquisitive reconciliation

There are two prominent treatments of disjunction in formal semantics. Traditionally, disjunction is taken to express an operator that applies to any two elements A and B of a Boolean algebra and yields their join. In particular, if A and B are propositions, then disjunction delivers their union, . Another, more recent proposal is to treat disjunction as expressing an operator that can apply to any two objects of the same semantic type, and yields the set consisting of these two objects. In particular, if disjunction applies to two propositions A and B, it delivers a set of propositional alternatives, . Each of the two approaches has certain merits that the other one lacks. Thus, it would be desirable to reconcile the two, combining their respective strengths. This paper shows that this is indeed possible, if we adopt a notion of meaning that does not just take truth-conditional, informative content into consideration, but also inquisitive content.