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Relative Necessity and Propositional Quantification


Following Smiley’s (The Journal of Symbolic Logic, 28, 113–134 1963) influential proposal, it has become standard practice to characterise notions of relative necessity in terms of simple strict conditionals. However, Humberstone (Reports on Mathematical Logic, 13, 33–42 1981) and others have highlighted various flaws with Smiley’s now standard account of relative necessity. In their recent article, Hale and Leech (Journal of Philosophical Logic, 46, 1–26 2017) propose a novel account of relative necessity designed to overcome the problems facing the standard account. Nevertheless, the current article argues that Hale & Leech’s account suffers from its own defects, some of which Hale & Leech are aware of but underplay. To supplement this criticism, the article offers an alternative account of relative necessity which overcomes these defects. This alternative account is developed in a quantified modal propositional logic and is shown model-theoretically to meet several desiderata of an account of relative necessity.


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For helpful discussion and feedback, thanks are due to Arif Ahmed, Robbie Williams, Timothy Williamson, and an audience at the Twelfth Annual Cambridge Graduate Conference on the Philosophy of Mathematics and Logic, at which Arif Ahmed provided comments on the paper. Special thanks are due to James Studd for particularly helpful feedback on earlier drafts, and to two anonymous referees for this journal whose comments greatly improved the paper.

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Correspondence to Alexander Roberts.

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Roberts, A. Relative Necessity and Propositional Quantification. J Philos Logic 49, 703–726 (2020).

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  • Relative necessity
  • Absolute necessity
  • Logical necessity
  • Modal logic