Inquisitive logic is an extension of classical logic which can express questions. To enable this expressiveness, a possible world semantics is used. So, it is natural to combine inquisitive and modal logic, thus obtaining the inquisitive modal logic \(\textrm{InqML}\) . This paper contributes to the model theory of \(\textrm{InqML}\) . We show that the filtration technique can be adapted to the inquisitive logic semantics. Using filtrations, we prove that \(\textrm{InqML}\) has the finite model property, i.e., every satisfiable formula is satisfiable in a finite model. As a consequence, we obtain that \(\textrm{InqML}\) is decidable. Finally, we study filtrations over particular model classes and we prove that some extensions of \(\textrm{InqML}\) are also decidable.
