May 2021 Modal Metatheory for Quantified Modal Logic, With and Without the Barcan Formulas
Andrew Joseph McCarthy
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Notre Dame J. Formal Logic 62(2): 285-301 (May 2021). DOI: 10.1215/00294527-2021-0015


This paper develops some modal metatheory for quantified modal logic. In such a theory, the logic of a first-order modal object-language is made sensitive to the modal facts, stated in the metalanguage. This is radically different from possible worlds semantics, which reduces questions of validity to questions of nonmodal set theory. We consider theories which characterize a notion of truth under a second-order interpretation, where an operator for metaphysical necessity is treated homophonically. The form they take is crucially influenced by whether the Barcan formulas are part of the metalogic. The main result is that, with the Barcan formulas, formulating a modal metatheory is easy; principles of classical semantics transfer over smoothly with natural adaptations to the modal setting. Without the Barcan formulas, it seems to be much harder. Several options are considered, but each involves commitments plausibly in tension with the underlying philosophical motivations for rejecting them.


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Andrew Joseph McCarthy. "Modal Metatheory for Quantified Modal Logic, With and Without the Barcan Formulas." Notre Dame J. Formal Logic 62 (2) 285 - 301, May 2021.


Received: 21 April 2020; Accepted: 10 November 2020; Published: May 2021
First available in Project Euclid: 9 June 2021

Digital Object Identifier: 10.1215/00294527-2021-0015

Primary: 03B45

Keywords: Barcan formulas , contingentism , necessitism , quantified modal logic

Rights: Copyright © 2021 University of Notre Dame


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Vol.62 • No. 2 • May 2021
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