  In the first part of this paper we analyzed finite non-deterministic matrix
semantics for propositional non-normal modal logics as an alternative to the
standard Kripke's possible world semantics. This kind of modal systems
characterized by finite non-deterministic matrices was originally proposed by
Ju. Ivlev in the 70's. The aim of this second paper is to introduce a formal
non-deterministic semantical framework for the quantified versions of some
Ivlev-like non-normal modal logics. It will be shown that several well-known
controversial issues of quantified modal logics, relative to the identity
predicate, Barcan's formulas, and de dicto and de re modalities, can be tackled
from a new angle within the present framework.
