Kripke Completeness for Predicate Modal Logics of Bounded Alternativity: First Steps

We study Kripke completeness for predicate modal logics with propositional axioms of bounded alternativity corresponding to the condition on Kripke frames that every world sees at most n worlds. We prove that minimal predicate modal logics with axioms of bounded alternativity, as well as their extensions with the axiom of reflexivity, are strongly Kripke complete, whereas the addition of the axiom of transitivity results in Kripke incomplete logics, both with and without reflexivity. We show that these incomplete logics are members of an uncountable family of Kripke incomplete logics and give explicit axiomatizations of completions for some of them.