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Syntactic characterizations of \u1D4B-completeness and \u1D4A\u1D4F-completeness

Develop necessary and sufficient syntactic characterizations for when a normal modal logic is \u1D4B-complete (complete additivity) or \u1D4A\u1D4F-complete (complete and completely additive), analogous to the known characterization of \u1D4C-completeness via conservativity of the minimal tense extension.

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Background

The paper recalls a precise syntactic condition characterizing \u1D4C-completeness (tense algebras) via conservativity of the minimal tense extension. In contrast, comparable characterizations for \u1D4B- and \u1D4A\u1D4F-completeness are not known.

Establishing such characterizations would align algebraic and syntactic perspectives on completeness in possibility semantics, illuminating the boundaries of principal and full possibility frame completeness beyond the \u1D4C-case.

References

Characterizations of \u1D4B-completeness and \u1D4A\u1D4F-completeness in terms of conservativity of extensions are open questions.

— Possibility Frames and Forcing for Modal Logic (2501.11768 - Holliday, 20 Jan 2025) in Section Completeness for Principal Possibility Frames (\S~\ref{SyntacticProp})