Skip to main content
Download PDF

Roads to Necessitarianism

Journal of Philosophical Logic volume 50, pages 89–96 (2021)Cite this article

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.

References

  1. Arló-Costa, H. (2001). Bayesian epistemology and epistemic conditionals: on the status of the export-import laws. The Journal of Philosophy, 98(11), 555–593.

    Google Scholar 

  2. Fine, K. (2012). Counterfactuals without possible worlds. The Journal of Philosophy, 109, 221–246.

    Article  Google Scholar 

  3. Fine, K. (2017). Truthmaker semantics. In Wrigh, B.H.C., & Miller, A. (Eds.) Blackwell companion to the philosophy of language: Blackwell.

  4. Gibbard, A. (1981). Two recent theories of conditionals. In Harper, W.L., Stalnaker, R., & Pearce, G. (Eds.) Ifs: conditionals, beliefs, decision, chance, and time (pp. 211–247). Dordrecht: Reidel.

  5. Gillies, A. (2009). On truth conditions for If (but not quite only If). Philosophical Review, 118(3), 325–349.

    Article  Google Scholar 

  6. Hakli, R., & Negri, S. (2011). Does the deduction theorem fail for modal logic? Synthese, 187(3), 849–867. https://doi.org/10.1007/s11229-011-9905-9.

    Article  Google Scholar 

  7. Khoo, J. (2013). A note on Gibbard’s proof. Philosophical Studies, 166(1), 153–164.

    Article  Google Scholar 

  8. Khoo, J., & Mandelkern, M. (2019). Triviality results and the relationship between logical and natural languages. Mind, 128(510), 485–526. https://doi.org/10.1093/mind/fzy006.

    Article  Google Scholar 

  9. Kratzer, A. (1981). The notional category of modality. In Eikmeyer, H., & Rieser, H. (Eds.) Words, worlds, and contexts: new approaches in word semantics (pp. 38–74): de Gruyter.

  10. Kratzer, A. (1986). Conditionals. Chicago Linguistics Society, 22(2), 1–15.

    Google Scholar 

  11. Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.

    Google Scholar 

  12. Mandelkern, M. (2019). Crises of Identity. In Schlöder, J.J., McHugh, D., & Roelofsen, F. (Eds.) Proceedings of the 22nd Amsterdam Colloquium (pp. 279–288). University of Amsterdam.

  13. Mandelkern, M. (2020). If p, then p! Manuscript, All Souls College, Oxford. https://philpapers.org/rec/MANIPT-2.

  14. Mandelkern, M. (2020). Import-export and ‘and’. Philosophy and Phenomenological Research, 100(1), 118–135. https://doi.org/10.1111/phpr.12513.

    Article  Google Scholar 

  15. McGee, V. (1985). A counterexample to modus ponens. The Journal of Philosophy, 82(9), 462–471.

    Article  Google Scholar 

  16. Ramsey, F.P. (1978/1931). General propositions and causality. In Mellor, D.H. (Ed.) Foundations: essays in philosophy, logic, mathematics and economics (pp. 133–51): Routledge and Kegan Paul.

  17. Stalnaker, R. (1968). A theory of conditionals. In Rescher, N. (Ed.) Studies in logical theory (pp. 98–112). Oxford: Blackwell.

  18. Stalnaker, R.C., & Thomason, R.H. (1970). A semantic analysis of conditional logic. Theoria, 36(1), 23–42. https://doi.org/10.1111/j.1755-2567.1970.tb00408.x.

    Article  Google Scholar 

  19. van Wijnbergen-Huitink, J., Elqayam, S., & Over, D.E. (2014). The probability of iterated conditionals. Cognitive Science, 1–16. https://doi.org/10.1111/cogs.12169.

  20. Yalcin, S. (2007). Epistemic modals. Mind, 116(464), 983–1026. https://doi.org/10.1093/mind/fzm983.

    Article  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. All Souls College, Oxford, OX1 4AL, UK

    Matthew Mandelkern

  2. UCL, London, UK

    Daniel Rothschild

Authors
  1. Matthew Mandelkern
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Rothschild
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Mandelkern.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

    ⊧(¬p ∧ p) > ¬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 p⊧q then (p ∧ r)⊧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.. ⊧(¬p ∧ p) > ¬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

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-020-09562-9

Keywords

  • Logic of conditionals
  • Triviality results