In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order to embody both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals.
