November 2022 Modal Logics That Are Both Monotone and Antitone: Makinson’s Extension Results and Affinities between Logics
Lloyd Humberstone, Steven T. Kuhn
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Notre Dame J. Formal Logic 63(4): 515-550 (November 2022). DOI: 10.1215/00294527-2022-0029


A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV, or LF, and every antitone logic can be extended to LN, LV, or LF, where LI, LN, LV, and LF are logics axiomatized, respectively, by the schemas αα, α¬α, α, and α. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF, and the minimum amphitone logic AM axiomatized by the schema αβ. These logics, along with LI, LN, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over.


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Lloyd Humberstone. Steven T. Kuhn. "Modal Logics That Are Both Monotone and Antitone: Makinson’s Extension Results and Affinities between Logics." Notre Dame J. Formal Logic 63 (4) 515 - 550, November 2022.


Received: 25 March 2021; Accepted: 14 August 2022; Published: November 2022
First available in Project Euclid: 16 December 2022

MathSciNet: MR4522325
zbMATH: 07634481
Digital Object Identifier: 10.1215/00294527-2022-0029

Primary: 03B45
Secondary: 03A05 , 03B05 , 03F25

Keywords: classical propositional logic , consequence relations , equivalence between logics , extensional logics , Makinson , modal logic , notational variance , translational embeddings , translations

Rights: Copyright © 2022 University of Notre Dame


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Vol.63 • No. 4 • November 2022
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