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Roads to Necessitarianism

Abstract

We show that each of three natural sets of assumptions about the conditional entails necessitarianism: that anything possible is necessary. Since most agree that this conclusion is obviously false, this shows that at least one member of each set of assumptions must be rejected. All of these assumptions are, however, widely accepted and well-motivated. This creates a puzzle which we leave open.

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Acknowledgments

Many thanks to two anonymous referees for this journal and to Angelika Kratzer for helpful feedback on this paper.

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Correspondence to Matthew Mandelkern.

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Appendix

Appendix

A.1 First Proof: Right Monotonicity

For arbitrary p:

  1. i.

    \(\models \Box (\neg (\neg p \wedge p))\supset { ((\neg p \wedge p)>\neg p)}\) Conditional Quodlibet

  2. ii.

    \( \Box \neg (\neg p \wedge p) \models { (\neg p \wedge p)>\neg p}\) Deduction Theorem, (i)

  3. iii.

    ⊧(¬pp) > ¬p K, (ii)

  4. iv.

    ⊧¬p > (p > ¬p) Import-Export, (iii)

  5. v.

    \(\models \lozenge p \supset \neg (p>\neg p )\) Restricted Aristotle’s Thesis, PC, K

  6. vi.

    \(\models \lozenge p \supset (\neg (p>\neg p )\vee p)\) PC, (v)

  7. vii.

    \(\models \lozenge p\supset ((p>\neg p )\supset p)\) PC, (vi)

  8. viii.

    \(\lozenge p\models (p>\neg p )\supset p\) Deduction Theorem, (vii)

  9. ix.

    \(\Box \lozenge p\models \neg p> p\) Right Monotonicity, (iv), (viii)

  10. x.

    \(\Box \lozenge p\models \Box p\) Restricted Aristotle’s Thesis, PC, (ix)

  11. xi.

    \(\lozenge p\models \Box \lozenge p\) 5

  12. xii.

    \(\lozenge p\models \Box p\) PC, (x), (xi)

A.2 Second Proof: Logical Implication and Logical Ad Falsum

We note first that the following is entailed by our definition of entailment and our classical semantics for ‘∧’:

  • Monotonicity: If pq then (pr)⊧q

Then, for arbitrary p:

  • i.   \(\models \Box (\neg ((\lozenge p\wedge \neg p )\wedge p))\supset (({(\lozenge p\wedge \neg p) \wedge p)>\neg p)}\)

  •        Conditional Quodlibet

  • ii.   \(\Box \neg ((\lozenge p\wedge \neg p) \wedge p)\models ({(\lozenge p\wedge \neg p) \wedge p)>\neg p}\)

  •        Deduction Theorem, (i)

  • iii.  \( \models ((\lozenge p\wedge \neg p) \wedge p)>\neg p\)    K, (ii)

  • iv.    \(\models {(\lozenge p \wedge \neg p) > (p>\neg p) }\)    Import-Export, (iii)

  • v.     \(\models \lozenge p \supset \neg (p>\neg p )\)     Restricted Aristotle’s Thesis, PC, K

  • vi.     \(\lozenge p \models \neg (p>\neg p ) \)     Deduction Theorem, (v)

  • vii.     \(\lozenge p\wedge \neg p \models \neg (p>\neg p )\)     Monotonicity, (vi)

  • viii.     \(\models (\lozenge p\wedge \neg p)> \neg (p>\neg p )\)    Logical Implication, (vii)

  • ix.     \(\models \neg (\lozenge p\wedge \neg p)\)     Logical Ad Falsum, (iv), (viii)

  • x.     \(\models \lozenge p \supset p\)     PC, (ix)

  • xi.     \(\models \Box \lozenge p \supset \Box p\)     K, (x)

  • xii.     \(\models \lozenge p\supset \Box \lozenge p\)     5

  • xiii.     \(\models \lozenge p \supset \Box p\)     PC, (xi), (xii)

  • xiv.     \(\lozenge p \models \Box p\)     Deduction Theorem, (xiii)

A.3 Third Proof: Nothing Added

For arbitrary p:

  • i. \(\models \Box (\neg (\neg p \wedge p))\supset {((\neg p \wedge p)>\neg p)}\) Conditional Quodlibet

  • ii. \( \Box \neg (\neg p \wedge p)\models {(\neg p \wedge p)>\neg p}\) Deduction Theorem, (i)

  • iii.. ⊧(¬pp) > ¬p K, (ii)

  • iv. ⊧¬p > (p > ¬p) Import-Export, (iii)

  • v. \(\models \lozenge p \supset \neg (p>\neg p )\) Restricted Aristotle’s Thesis, PC, K

  • vi \(\models \Box (\lozenge p \supset \neg (p>\neg p ))\) K, (v)

  • vii. \(\models \Box \lozenge p \supset \Box \neg (p>\neg p )\) K, (vi)

  • viii. \(\models \lozenge p\supset \Box \lozenge p\) 5

  • ix. \(\models \lozenge p \supset \Box \neg (p>\neg p )\) PC, (vii), (viii)

  • x. \(\lozenge p \models \Box \neg (p>\neg p )\) Deduction Theorem, (ix)

  • xi. \(\lozenge p \models \Box \neg (p>\neg p ) \supset ((p>\neg p)>(\neg p>p))\)

  • Conditional Quodlibet

  • xii. \(\lozenge p \models {(p>\neg p)>(\neg p>p)}\) PC, (x), (xi)

  • xiii. \(\lozenge p \models {((p>\neg p)\wedge \neg p)>p}\) (xii), Import-Export

  • xiv. \(\lozenge p \models {(\neg p \wedge (p>\neg p))>p}\) (xiii), PC

  • xv. \(\lozenge p \models {\neg p > ((p>\neg p)>p)}\) (xiv), Import-Export

  • xvi. \(\lozenge p \models (\neg p > ((p>\neg p)>p)) \equiv ({\neg p> p})\) (iv), Nothing Added

  • xvii. \(\lozenge p \models {\neg p> p}\) (xv), (xvi), PC

  • xviii. \(\lozenge p \models \Box p\) Restricted Aristotle’s Thesis, (xvii), PC

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Mandelkern, M., Rothschild, D. Roads to Necessitarianism. J Philos Logic 50, 89–96 (2021). https://doi.org/10.1007/s10992-020-09562-9

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Keywords

  • Logic of conditionals
  • Triviality results