As well as considering modalities applied to propositions in propositionalmodal logic, logicians have also studied modalities applied to predicate logic, or first-order modal logic. Here we can represent statements such as
, Necessarily there exists something which is .
, There exists something which is necessarily .
In the possible worlds setting, we think of a world containing a collection of individuals, instantiating various relations. Then we may require that for an individual in one world, for every related world there be a counterpart there. See Kracht & Kutz.
Various axioms concerning the interaction of the quantifiers and modal oeprators have been suggested, such as the Barcan formula
\Box \forall x F x \to \forall x \Box F x,
but this is not considered to hold generally.
Awodey and Kishida showed that S4 first-order modal logic is complete with respect to a sheaf-theoretic semantics. Other approaches involve modal metaframes (Shehtman/Skvortsov 1993), counterpart frames (Kracht/Kutz 2002), coherence frames (Kracht/Kutz 2005), general metaframes (Shirasu 1998) and ionads?.
Steve Awodey and Kohei Kishida?, Topology and Modality: The Topological Interpretation of First-Order Modal Logic, (pdf)
Marcus Kracht and Oliver Kutz, Logically Possible Worlds and Counterpart Semantics for Modal Logic,(pdf)
Kracht/Kutz 2002, The Semantics of Modal Predicate Logic I. Counterpart–Frames., Advances in Modal Logic. Volume 3
Kracht/Kutz 2005, The Semantics of Modal Predicate Logic II. Modal Individuals Revisited, ASL Lecture Notes in Logic, vol. 22, pp. 60–97. (pdf.
Shirasu 1998, Hiroyuki Shirasu, Duality in super-intuitionistic and modal predicate logics,
Shehtman/Skvortsov 1993, Maximal Kripke–Type Semantics for Modal and Superintuitionistic Predicate Logics, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 69–101.