Axiomatizability of LΠ-logics for neighborhood possibility frames

Determine whether the LΠ-logic of each of the following classes of neighborhood possibility frames is recursively axiomatizable or at least recursively enumerable: (i) all neighborhood possibility systems (equivalently, complete interior algebras), (ii) all basic neighborhood possibility frames, (iii) all normal neighborhood possibility frames in which each neighborhood N(x) is a proper filter, and (iv) other natural classes of neighborhood possibility frames. The objective is to clarify the proof-theoretic status of propositional quantification over modalities within possibility semantics beyond the cases already settled.

Background

In the neighborhood-based strand of possibility semantics, propositional quantification (the LΠ-language) is interpreted over Boolean algebras of regular open sets with neighborhood operators. While several completeness results are known—e.g., KD4^∀5Π over classes corresponding to proper filter algebras—general axiomatizations for broader frame classes remain unsettled.

The text explicitly notes an open status regarding whether logics of propositional quantification for various neighborhood possibility frame classes (including the full class of neighborhood possibility systems and normal frames with proper filter neighborhoods) admit recursive axiomatizations or at least enumerations.