David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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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.
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