Bounded Inquisitive Logics: Sequent Calculi and Schematic Validity

Propositional inquisitive logic is the limit of its $n$-bounded approximations. In the predicate setting, however, this does not hold anymore, as discovered by Ciardelli and Grilletti, who also found complete axiomatizations of $n$-bounded inquisitive logics ${\mathsf{I}\mathsf{n}\mathsf{q}\mathsf{B}\mathsf{Q}}_n$, for every fixed $n$. We introduce cut-free labelled sequent calculi for these logics. We illustrate the intricacies of $\mathit{\text{schematic}}\mathit{\text{validity}}$ in such systems by showing that the well-known Casari formula is $\mathit{\text{atomically}}$ valid in (a weak sublogic of) predicate inquisitive logic $\mathsf{I}\mathsf{n}\mathsf{q}\mathsf{B}\mathsf{Q}$, fails to be schematically valid in it, and yet is schematically valid under the finite boundedness assumption. The derivations in our calculi, however, are guaranteed to be schematically valid whenever a single specific rule is not used.