Algebraic foundations for inquisitive semantics

Traditionally, meaning is identified with informative content. The central aim of inquisitive semantics [1,2,4,5, a.o.] is to develop a notion of semantic meaning that embodies both informative and inquisitive content. To achieve this, the proposition expressed by a sentence ϕ, [ϕ], is not taken to be a set of possible worlds, but rather a set of possibilities, where each possibility in turn is a set of possible worlds. In uttering a sentence ϕ, a speaker provides the information that the actual world is contained in at least one possibility in [ϕ], and at the same time she requests enough information from other participants to establish for at least one possibility α ∈ [ϕ] that the actual world is contained in α.