Modal Logics Between Propositional and First Order

One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery produces a decidable logic, but adding it to S5 produces an undecidable one. Further, if an equality symbol is in the language, and interpreted by the equality relation, logics from K4 to S5 yield undecidable versions. (Thus transitivity is the villain here.) The proof of undecidability consists in showing that classical first-order logic can be embedded.
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Melvin Fitting (2004). First-Order Intensional Logic. Annals of Pure and Applied Logic 127 (1-3):171-193.
Melvin Fitting (2006). FOIL Axiomatized. Studia Logica 84 (1):1 - 22.

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