Full Idea
If the domain of propositional quantification is constant across worlds, the Barcan formula and its converse hold.
Gist of Idea
If the domain of propositional quantification is constant, the Barcan formulas hold
Source
Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2)
Book Reference
-: 'Dialectica' [-], p.259
A Reaction
So the issue is whether we should take metaphysics to be dealing with a constant or varying domains. Williamson seems to favour the former, but my instincts incline towards the latter.
Related Ideas
Idea 15132 The Barcan formulas fail in models with varying domains [Kripke, by Williamson]
Idea 9449 The plausible Barcan formula implies modality in the actual world [Bird]