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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 Ref
-: '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]
| 15131 | If metaphysical possibility is not a contingent matter, then S5 seems to suit it best [Williamson] |
| 15133 | A thing can't be the only necessary existent, because its singleton set would be as well [Williamson] |
| 15134 | The truthmaker principle requires some specific named thing to make the difference [Williamson] |
| 15135 | If the domain of propositional quantification is constant, the Barcan formulas hold [Williamson] |
| 15138 | Not all quantification is objectual or substitutional [Williamson] |
| 15136 | Substitutional quantification is metaphysical neutral, and equivalent to a disjunction of instances [Williamson] |
| 15137 | If 'fact' is a noun, can we name the fact that dogs bark 'Mary'? [Williamson] |
| 15142 | Our ability to count objects across possibilities favours the Barcan formulas [Williamson] |
| 15139 | Converse Barcan: could something fail to meet a condition, if everything meets that condition? [Williamson] |
| 15141 | Truthmaker is incompatible with modal semantics of varying domains [Williamson] |
| 15140 | The converse Barcan formula will not allow contingent truths to have truthmakers [Williamson] |
| 18492 | Not all quantification is either objectual or substitutional [Williamson] |