A Misleading Triviality Argument in The Theory of Conditionals
DOI:
https://doi.org/10.12775/LLP.2024.015Keywords
probability of conditionals, Dutch Book, triviality results, Hájek’s update rules, PCCPAbstract
PCCP is the much discussed claim that the probability of a conditional A → B is conditional probability. Triviality results purport to show that PCCP – as a general claim – is false. A particularly interesting proof has been presented in (Hájek, 2011), who shows that – even if a probability distribution P initially satisfied PCCP – a rational update can produce a non-PCCP probability distribution.
We argue that the notion of rational update in this argumentation is construed in much too broad a way. In order to make the argumentation precise, we discuss the general rules for modeling conditionals in probability spaces and present formalized version(s) of PCCP and of minimal assumptions concerning the appropriate spaces. Using the introduced apparatus we give a detailed analysis of Hájek’s (2011) triviality proof and show that it is based on an application of revision rules which allow one to construct probability distributions violating not only PCCP, but also fundamental properties of conditionals.
This means that they do not really provide arguments against PCCP, properly formalized. We also discuss a Dutch Book argument which shows that the updated belief system is not coherent. This gives an additional, strong argument against accepting the update rules. We also discuss the Converse Dutch Book theorem and argue, that even if the produced probability measure seems to violate it, it cannot serve as the counterexample, as it is not an appropriate model for conditionals. Ultimately, we show that important arguments against PCCP fail.
References
Adams, E. W., 1965, “On the logic of conditionals”, Inquiry 8: 166–197. DOI: : https://doi.org/10.1080/00201746508601430
Adams, E. W., 1970, “Subjunctive and indicative conditionals”, Foundations of Language 6: 89–94. DOI: : https://doi.org/10.2307/2272204
Adams, E. W., 1975,The Logic of Conditionals, Dordrecht: D. Reidel.
Adams, E. W., 1998, “A useful four-valued logic”, pages 7–37 in J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, Boston: Reidel Publishing Company. DOI: : https://doi.org/10.1007/978-94-010-1161-7_2
Bennett, J., 2003, A Philosophical Guide to Conditionals, Oxford: Clarendon Press.
Berto, R., and A. Özgün, 2021, “Indicative conditionals: Probabilities and relevance”, Philosophical Studies 178: 3697–3730. DOI: : https://doi.org/s11098-021-01622-3
Cruz, N., D. Over, M. Oaksford, and J. Baratgin, 2016, “Centering and the meaning of conditionals”, pages 1104–1109 in A. Papafragou, D. Grodner, D. Mirman, and J. C. Trueswell (eds.), Proceedings of the 38th Annual Conference of the Cognitive Science Society, Cognitive Science Society.
de Finetti, B., 1937, “La Prévision: ses lois logiques, ses sources subjectives”, Annales de l’Institut Henri Poincaré 7: 1–68; trans. as “Foresight: Its Logical Laws, its Subjective Sources”, pages 93–159 in H. E. Kyburg and H. E. Smokler (eds.), Studies in Subjective Probability, New York: Wiley, 1980.
Easwaran, K., 2011a, “Bayesianism I: Introduction and arguments in favor”, Philosophy Compass 6(5): 312–320. DOI: : https://doi.org/10.1111/j.1747-9991.2011.00399.x
Easwaran, K., 2011b, “Bayesianism II: Criticisms and applications”, Philosophy Compass 6(5): 321–332. DOI: : https://doi.org/10.1111/j.1747-9991.2011.00398.x
Edgington, D., 1995, “On conditionals”, Mind 104: 235–329. DOI: : https://doi.org/10.1093/mind/104.414.235
Edgington, D., 2020, “Indicative conditionals”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2020 Edition), : https://plato.stanford.edu/archives/fall2020/entries/conditionals/
Egré, P., and H. Rott, 2021, “The logic of conditionals”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter 2021 Edition). : https://plato.stanford.edu/archives/win2021/entries/logic-conditionals/
Evans, J. St. B. T., and D. E. Over, 2004, If: Supposition, Pragmatics, and Dual Processes, Oxford: Oxford University Press.
Fitelson, B., 2015, “The strongest possible Lewisian triviality result”, Thought 4: 69–74. DOI: : https://doi.org/10.1002/tht3.159
Grice, H. P., 1989, Studies in the Way of Words, Cambridge: Harvard University Press.
Hájek, A., 1989, “Probabilities of conditionals revisited”, Journal of Philosophical Logic 18: 423–428.
Hájek, A., 2008, “Arguments for – or against – probabilism?”, British Journal for the Philsophy of Science 59(4): 793–819. DOI: : https://doi.org/10.1093/bjps/axn045
Hájek, A., 2009, “Dutch Book arguments”, pages 173–195 in P. Anand, P. K. Pattanaik, C. Puppe (eds.),The Handbook of rational and social choice, Oxford University Press. DOI: : https://doi.org/10.1093/acprof:oso/9780199290420.001.0001
Hájek, A., 2011,“Triviality pursuit”, Topoi 30(1): 3–15. DOI: : https://doi.org/10.1007/s11245-010-9083-2
Hájek, A., 2012, “The Fall of “Adams’ Thesis”?”, Journal of Logic, Language and Information 21: 145–161. DOI: : https://doi.org/10.1007/s10849-012-9157-1
Hájek, A., and N. Hall, 1994, “The hypothesis of the conditional construal of conditional probability”, pages: 75–110 in E. Eells and B. Skyrms (eds.), Probabilities and Conditionals: Belief Revision and Rational Decision, Cambridge: Cambridge University Press.
Hall, N., 1994, “Back in the (CCCP)”, pages 141–160 in E. Eells and B. Skyrms (eds.), Probabilities and Conditionals: Belief Revision and Rational Decision, Cambridge: Cambridge University Press.
Jackson, F., 1987, Conditionals, New York: Blackwell.
Kaufmann, S., 2004, “Conditioning against the grain: Abduction and indicative conditionals”, Journal of Philosophical Logic 33(6): 583–606. DOI: : https://doi.org/10.1023/B:LOGI.0000046142.51136.bf
Kaufmann, S. 2005, “Conditional predictions: A probabilistic account”, Linguistics and Philosophy 28(2): 181–231. DOI: : https://doi.org/10.1007/s10988-005-3731-9
Kaufmann, S., 2009, “Conditionals right and left: Probabilities for the whole family”, Journal of Philosophical Logic 38: 1–53. DOI: : https://doi.org/10.1007/s10992-008-9088-0
Kaufmann, S., 2015, “Conditionals, conditional probability, and conditionalization”, pages 71–94 in H. Zeevat and H. C. Schmitz (eds.), Bayesian Natural Language Semantics and Pragmatics, New York, Heidelberg: Springer.
Kaufmann, S., 2023, “Bernoulli semantics and ordinal semantics for conditionals”, Journal of Philosophical Logic 52: 199–220. DOI: : https://doi.org/10.1007/s10992-022-09670-8
Khoo, J., and P. Santorio, 2018, Lecture Notes: Probability of Conditionals in Modal Semantics. : http://paolosantorio.net/teaching.html
Lewis, D., 1976, “Probabilities of conditionals and conditional probabilities”, Philosophical Review 85: 297–315. DOI: : https://doi.org/10.2307/2184045
Lewis, D., 1986, “Probabilities of conditionals and conditional rrobabilities II”, Philosophical Review 95: 581–589. DOI: : https://doi.org/10.2307/2185051
McGee, V., 1989, “Conditional probabilities and compounds of conditionals”, Philosophical Review 98(4): 485–541. DOI: : https://doi.org/10.2307/2185116
Milne, P., 2003, “The simplest Lewis-style triviality proof yet?”, Analysis 63(4): 300–303. DOI: : https://doi.org/10.1093/analys/63.4.300
Ramsey, F. P., 1990, “General propositions and causality”, pages 145–163 in D. H. Mellor (ed.), Philosophical Papers, Cambridge: Cambridge University Press.
Rehder, W., 1982, “Conditions for probabilities of conditionals to be conditional probabilities”, Synthese 53: 439–443.
Stalnaker, R., 1968, “A theory of conditionals”, Studies in Logical Theory, American Philosophical Quarterly 2: 98–112.
Stalnaker, R., 2009, “Conditional propositions and conditional assertions”, pages 277–248 in A. Egan and B. Weatherson (eds.), Epistemic Modality, Oxford: Oxford University Press. DOI: : https://doi.org/10.1093/acprof:oso/9780199591596.003.0008
Stalnaker, R., 2019, Knowledge and Conditionals, Oxford: Oxford University Press.
van Fraassen, B. C., 1976, “Probabilities of conditionals”, pages 261–308 in W. L. Harper, R. Stalnaker, and G. Pearce (eds.), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, Dordrecht: D. Reidel.
van Fraassen, B. C., 1984, “Belief and the will”, The Journal of Philosophy 81(5): 235–256. DOI: : https://doi.org/10.2307/2026388
Vineberg, S., 2016, “Dutch Book arguments”, in E. N. Zalta (ed.),The Stanford Encyclopedia of Philosophy (Spring 2016 Edition). : https://plato.stanford.edu/archives/spr2016/entries/dutch-book/
Węgrecki, J., and L. Wroński, 2023, “Revisiting the conditional construal of conditional probability”, Logic and Logical Philosophy 32(2): 261–268. DOI: : https://doi.org/10.12775/LLP.2022.024
Wójtowicz, A., and K. Wójtowicz, 2021a, “A stochastic graphs semantics for conditionals”, Erkenntnis 86,: 1071–1105. DOI: : https://doi.org/10.1007/s10670-019-00144-z
Anna Wójtowicz, Krzysztof Wójtowicz Wójtowicz, A., and K. Wójtowicz, 2021b, “Dutch Book against Lewis”, Synthese 199: 9185–9217. DOI: : https://doi.org/10.1007/s11229-021-03199-0
Wójtowicz, K., and A. Wójtowicz, 2022, “A graph model for probabilities of nested conditionals”, Linguistics and Philosophy 45(3): 1–48. DOI: : https://doi.org/10.1007/s10988-021-09324-z
Wójtowicz, A., and K. Wójtowicz, 2023, “A minimal probability space for conditionals”, Journal of Philosophical Logic 52(3): 1385–1415. DOI: : https://doi.org/10.1007/s10992-023-09710-x
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Anna Wójtowicz, Krzysztof Wójtowicz
![Creative Commons License](http://i.creativecommons.org/l/by-nd/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 309
Number of citations: 0