Systematic identification of logics complete for possibility frames but incomplete for world frames

Identify and classify further congruential modal logics that are incomplete with respect to basic neighborhood world frames yet complete with respect to classes of basic neighborhood possibility frames. Develop criteria or constructions that systematically generate such logics.

Background

Beyond specific examples, the chapter calls for a broader map of logics whose semantics require possibility frames to achieve completeness. Such a classification would deepen understanding of the semantic strength gained by moving from worlds to possibilities.

This problem invites both model-theoretic and proof-theoretic analysis to characterize families of logics exhibiting this separation.