Let V = {p, p 1, p 2, p 3, ..., q, q 1, ...} be a countably infinite vocabulary of elementary letters1. A interpretation consists of an assignment of a truth value from true, false to each elementary letter in V. Each interpretation I is identified with the set of those elementary letters assigned true by I. Thus each interpretation is a subset of V.

This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).