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JSAI International Symposium on Artificial Intelligence

JSAI-isAI 2015: New Frontiers in Artificial Intelligence pp 3–14Cite as

Towards a Probabilistic Analysis for Conditionals and Unconditionals

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10091))

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.

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Notes

  1. 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. 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. 3.

    The typographical similarity with the conventions of Inquisitive Semantics [3, 8], is intended, although in this paper I do not pursue an in-depth study of this connection.

  4. 4.

    Certain counterexamples have been discussed in the literature, but there is hope that those can be explained as systematic deviations that do not undermine the general idea, but rather help to fine-tune its application in particular cases [14, 19, 29].

  5. 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. 6.

    Random variables with range \(\{0,1\}\) are also called indicator functions.

  7. 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. 8.

    See [10] for a survey of the status quo in 1994. My own attempt at a readable exposition is found in [15], but see [4] for more.

  9. 9.

    Van Fraassen called the construction “Stalnaker Bernoulli model” since he saw in it the probabilistic analog of a Stalnaker-style selection function.

  10. 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.

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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.

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Authors and Affiliations

  1. Department of Linguistics, University of Connecticut, 365 Fairfield Way, Unit 1145, Storrs Mansfield, CT, 06268, USA

    Stefan Kaufmann

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  1. Stefan Kaufmann
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Correspondence to Stefan Kaufmann .

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Editors and Affiliations

  1. Chiba University Graduate School of Engineering, Chiba, Japan

    Mihoko Otake

  2. University of Tsukuba, Tokyo, Japan

    Setsuya Kurahashi

  3. Fujitsu Laboratories, Kanagawa, Japan

    Yuiko Ota

  4. National Institute of Informatics, Tokyo, Japan

    Ken Satoh

  5. Ochanomizu University, Tokyo, Japan

    Daisuke Bekki

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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

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  • DOI: https://doi.org/10.1007/978-3-319-50953-2_1

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