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
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.