Abstract
The thesis that the probability of a conditional is the corresponding conditional probability of C, given A, enjoys wide currency among philosophers and growing empirical support in psychology. In this paper I ask how a probabilisitic account of conditionals along these lines could be extended to unconditional sentences, i.e., conditionals with interrogative antecedents. Such sentences are typically interpreted as equivalent to conjunctions of conditionals. This raises a number of challenges for a probabilistic account, chief among them the question of what the probability of a conjunction of conditionals should be. I offer an analysis which addresses these issues by extending the interpretation of conditonals in Bernoulli models to the case of unconditionals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
A competing usage in the descriptive linguistic literature calls the constituents protasis and apodosis, but this usage is not widespread in formal semantics or philosophy.
- 2.
This glosses over important points of variation, for instance as to whether the set of propositions is taken to be the set of possible answers, of true answers, and whether its members are required to cover or partition the set of all possibilities [9, 11, 13]. These are important issues, but they are not crucial for the purposes of this paper.
- 3.
- 4.
- 5.
\(\mathcal {F}\subseteq \wp (\varOmega )\) is \(\sigma \)-algebra iff it contains \(\varOmega \) and is closed under complement and countable union. \(\Pr : \mathcal {F}\mapsto [0,1]\) is a probability measure iff \(\Pr (\varOmega ) = 1\) and for any countable set of pairwise disjoint \(X_i \in \mathcal {F}, \Pr (\bigcup _i X_i) = \sum _i \Pr (X_i)\).
- 6.
Random variables with range \(\{0,1\}\) are also called indicator functions.
- 7.
For continuous variables, the summation is replaced by integration, but the basic idea is the same. I write ‘\(Pr(\zeta =x)\)’ as shorthand for ‘\(Pr\left( \{\omega \in \varOmega |\zeta (\omega )=x\}\right) \)’.
- 8.
- 9.
Van Fraassen called the construction “Stalnaker Bernoulli model” since he saw in it the probabilistic analog of a Stalnaker-style selection function.
- 10.
More complex compounds also receive truth values and probabilities under the approach, but those are hard to evaluate because intuitive judgments are not easy to come by.
References
Adams, E.: The logic of conditionals. Inquiry 8, 166–197 (1965)
Bennett, J.: A Philosophical Guide to Conditionals. Oxford University Press, Oxford (2003)
Ciardelli, I., Roelofsen, F.: Generalized inquisitive semantics and logic (2009). http://sites.google.com/site/inquisitivesemantics/. Accessed Nov 2009
Edgington, D.: On conditionals. Mind 104(414), 235–329 (1995)
Eells, E., Skyrms, B. (eds.): Probabilities and Conditionals: Belief Revision and Rational Decision. Cambridge University Press, Cambridge (1994)
Evans, J.S., Over, D.E.: If. Oxford University Press, Oxford (2004)
van Fraassen, B.C.: Probabilities of conditionals. In: Harper, W.L., Stalnaker, R., Pearce, G. (eds.) Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. The University of Western Ontario Series in Philosophy of Science, vol. 1, pp. 261–308. D. Reidel, Dordrecht (1976)
Groenendijk, J., Roelofsen, F.: Inquisitive semantics and pragmatics (2009). http://sites.google.com/site/inquisitivesemantics/. Accessed Nov 2009
Groenendijk, J., Stokhof, M.: Studies in the semantics of questions and the pragmatics of answers. Ph.D. thesis, University of Amsterdam (1984)
Hájek, A., Hall, N.: The hypothesis of the conditional construal of conditional probability. In: Eells and Skyrms [5], pp. 75–110
Hamblin, C.L.: Questions. Australas. J. Philos. 36(3), 159–168 (1958)
Jeffrey, R.C.: If. J. Philos. 61, 702–703 (1964)
Karttunen, L.: Syntax and semantics of questions. Linguist. Philos. 1, 3–44 (1977)
Kaufmann, S.: Conditioning against the grain: abduction and indicative conditionals. J. Philos. Logic 33(6), 583–606 (2004)
Kaufmann, S.: Conditional predictions: a probabilistic account. Linguist. Philos. 28(2), 181–231 (2005)
Kaufmann, S.: Conditionals right and left: probabilities for the whole family. J. Philos. Logic 38, 1–53 (2009)
Kaufmann, S.: Unconditionals are conditionals. Handout, DIP Colloquium, University of Amsterdam. http://stefan-kaufmann.uconn.edu/Papers/Amsterdam2010_hout.pdf
Kaufmann, S.: Conditionals, conditional probabilities, and conditionalization. In: Zeevat, H., Schmitz, H.-C. (eds.) Bayesian Natural Language Semantics and Pragmatics. LCM, vol. 2, pp. 71–94. Springer, Cham (2015). doi:10.1007/978-3-319-17064-0_4
Khoo, J.: Probabilities of conditionals in context. Linguist. Philos. 39, 1–43 (2016)
Lewis, D.: Probabilities of conditionals and conditional probabilities. Philos. Rev. 85, 297–315 (1976)
Mellor, D.H. (ed.): Philosophical Papers: F. P. Ramsey. Cambridge University Press, New York (1990)
Oaksford, M., Chater, N.: Conditional probability and the cognitive science of conditional reasoning. Mind Lang. 18(4), 359–379 (2003)
Oaksford, M., Chater, N.: Bayesian Rationality: The Probabilistic Approach to Human Reasoning. Oxford University Press, Oxford (2007)
Ramsey, F.P.: General propositions and causality. Printed in [21], pp. 145–163 (1929)
Rawlins, K.: (Un)conditionals: an investigation in the syntax and semantics of conditional structures. Ph.D. thesis, UCSC (2008)
Rawlins, K.: (Un)conditionals. Nat. Lang. Seman. 21, 111–178 (2013)
Stalnaker, R.C.: A theory of conditionals. In: Harper, W.L., Stalnaker, R., Pearce, G. (eds.) IFS. Conditionals, Belief, Decision, Chance and Time. The Western Ontario Series in Philosophy of Science, vol. 15, pp. 41–55. Blackwell, Oxford (1968). doi:10.1007/978-94-009-9117-0_2
Stalnaker, R., Jeffrey, R.: Conditionals as random variables. In: Eells and Skyrms [5], pp. 31–46
Zhao, M.: Intervention and the probabilities of indicative conditionals. J. Philos. 112, 477–503 (2016)
Acknowledgments
I would like to thank the organizers of LENLS 12 for the opportunity to present this work. Parts of this material were previously presented at the “Work in Progress” seminar in the Philosophy Department at MIT. I am grateful to the audiences at both events for valuable feedback. Thanks also to Yukinori Takubo and Kyoto University for an invitation to a one-semester guest professorship in the fall of 2015, during which some of this work was carried out.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Kaufmann, S. (2017). Towards a Probabilistic Analysis for Conditionals and Unconditionals. In: Otake, M., Kurahashi, S., Ota, Y., Satoh, K., Bekki, D. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2015. Lecture Notes in Computer Science(), vol 10091. Springer, Cham. https://doi.org/10.1007/978-3-319-50953-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-50953-2_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50952-5
Online ISBN: 978-3-319-50953-2
eBook Packages: Computer ScienceComputer Science (R0)