We provide sound and complete axiomatizations for a large class of inquisitive modal logics. Inquisitive modal logic can be regarded as an extension of standard modal logic that allows to reason about informative and interrogative aspects of meaning in a uniform way. After introducing a suitable frame language for inquisitive modal logic, we construct modular labelled sequent calculi for all inquisitive modal logics characterized by a certain type of frame conditions called geometric implications. The construction is based on a general method developed by Negri. Each of our calculi is shown to satisfy cut-admissibility, height-preserving admissibility of weakening and contraction and height-preserving invertibility of all rules. The completeness of our proof systems is established by a countermodel construction in the style of Takeuti.
Valentin Müller (Mon,) studied this question.