A General Semantics for Quantified Modal Logic
by Edwin D. Mares and Robert Goldblatt
In [9] we developed a semantics for quantified relevant logic that uses
general frames. In this paper, we adapt that model theory to treat quantified
modal logics, giving a complete semantics to the quantified extensions, both
with and without the Barcan formula, of every propositional modal logic S.
If S is canonical our models are based on propositional frames that
validate S. We employ frames in which not every set of worlds is an admissible
proposition, and an alternative interpretation of the universal quantifier
using greatest lower bounds in the lattice of admissible propositions.
Our models have a fixed domain of individuals, even in the absence of the Barcan formula.
For systems with the Barcan formula it is possible to preserve the usual
Tarskian reading of the quantifier, at the expensive of sometimes losing
validity of S in the underlying propositional frames. We apply our results
to a number of logics, including S4.2, S4M and KW, whose quantified
extensions are incomplete for the standard semantics.