Kripke incompleteness of predicate extensions of the modal logics axiomatized by a canonical formula for a frame with a nontrivial cluster (Q1582312)

The author generalizes the Kripke incompleteness proof for predicate modal logic presented in \textit{G. E. Hughes} and \textit{M. J. Cresswell}'s book ``A new introduction to modal logic'' (Routledge, London) (1996; Zbl 0855.03002), and shows the incompleteness for some predicate modal logics in the extensions of S4 characterized by means of the frame formula for an arbitrarily given finite rooted Kripke frame with nontrivial cluster, in which some constant domain versions are included as well. In particular, the extension of first order (constant domain) S4 by the canonical formula associated to such a Kripke frame is shown to be Kripke incomplete. As a corollary the author obtains that for any subframe logic containing S4 its least constant domain first order extension is Kripke complete iff it has the finite embedding property.