Kripke completeness for predicate modal logics with axioms of bounded alternativity

Joint work with Valentin Shehtman.
We present some recent results on Kripke completeness, Kripke incompleteness, and Kripke completions of first-order modal logics with propositional axioms that, in propositional Kripke semantics, correspond to the condition that every world sees at most $n$ worlds. Kripke completeness is established using selective weak submodels of canonical models introduced by V. Shehtman. Kripke incompleteness is proven using the Kripke bundle semantics.