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
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.
Fine, K. (2012). Counterfactuals without possible worlds. The Journal of Philosophy, 109, 221–246.
Fine, K. (2017). Truthmaker semantics. In Wrigh, B.H.C., & Miller, A. (Eds.) Blackwell companion to the philosophy of language: Blackwell.
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.
Gillies, A. (2009). On truth conditions for If (but not quite only If). Philosophical Review, 118(3), 325–349.
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.
Khoo, J. (2013). A note on Gibbard’s proof. Philosophical Studies, 166(1), 153–164.
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.
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.
Kratzer, A. (1986). Conditionals. Chicago Linguistics Society, 22(2), 1–15.
Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.
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.
Mandelkern, M. (2020). If p, then p! Manuscript, All Souls College, Oxford. https://philpapers.org/rec/MANIPT-2.
Mandelkern, M. (2020). Import-export and ‘and’. Philosophy and Phenomenological Research, 100(1), 118–135. https://doi.org/10.1111/phpr.12513.
McGee, V. (1985). A counterexample to modus ponens. The Journal of Philosophy, 82(9), 462–471.
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.
Stalnaker, R. (1968). A theory of conditionals. In Rescher, N. (Ed.) Studies in logical theory (pp. 98–112). Oxford: Blackwell.
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.
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.
Yalcin, S. (2007). Epistemic modals. Mind, 116(464), 983–1026. https://doi.org/10.1093/mind/fzm983.
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
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 A.1 First Proof: Right Monotonicity
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.
\(\models \lozenge p \supset (\neg (p>\neg p )\vee p)\) PC, (v)
-
vii.
\(\models \lozenge p\supset ((p>\neg p )\supset p)\) PC, (vi)
-
viii.
\(\lozenge p\models (p>\neg p )\supset p\) Deduction Theorem, (vii)
-
ix.
\(\Box \lozenge p\models \neg p> p\) Right Monotonicity, (iv), (viii)
-
x.
\(\Box \lozenge p\models \Box p\) Restricted Aristotle’s Thesis, PC, (ix)
-
xi.
\(\lozenge p\models \Box \lozenge p\) 5
-
xii.
\(\lozenge p\models \Box p\) PC, (x), (xi)
1.2 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)
1.3 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/.
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-020-09562-9
Keywords
- Logic of conditionals
- Triviality results