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Learning from Simple Indicative Conditionals

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Abstract

An agent who receives information in the form of an indicative conditional statement and who trusts her source will modify her credences to bring them in line with the conditional. I will argue that the agent, upon the acquisition of such information, should, in general, expand her prior credence function to an indeterminate posterior one; that is, to a set of credence functions. Two different ways the agent might interpret the conditional will be presented, and the properties of the resulting indeterminate posteriors compared. The cause of the expansion from a single prior credence function to a set of credence functions forming the indeterminate posterior one will be explained. The expansion undermines the Bayesian dogma that the result of assimilating new information into a determinate prior credence functions is always a determinate posterior one.

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Notes

  1. This is intended to be an informal statement of what it means to learn something. I will not attempt to defend this definition in any way other than by stating that it, or something like it, is intuitively obvious.

  2. I am ignoring an important distinction here: if an agent is sure of some proposition, her credence in that proposition equals 1, but the opposite need not be true. When drawing a number from an infinite lottery, for example, the credence one has in drawing any particular number is 0, but that does not imply one should consider it impossible that that number will be drawn. The distinction is important only in Belief states with an infinite number of propositions, and it does not affect what I will say here about learning from conditionals.

  3. “¬” indicates negation, “\( \wedge \)” conjunction and “\( \vee \)” disjunction.

  4. I will treat the conclusion that, after acceptance of an indicative conditional, B cannot be false when A is true as uncontroversial. See also Edgington (1995 2.3) or Bennett (2003 p. 21).

  5. He does not explicitly state that his goal is a determinate posterior credence function, but it is clear from his article that he considers a complete theory of updating on simple indicative conditionals to be one that produces a determinate posterior Belief state.

  6. The logical symbol “\( \wedge \)” corresponds to the set-theoretic symbol “∩” for intersection. Logical negation corresponds to taking the complement, indicated by a superscript c added to the name of the complemented set. Consequently, the proposition A\( \wedge \neg \)B corresponds to the set A∩Bc.

  7. For example, it may contain worlds in which Unicorns exist. Their existence does not affect the truth of the indicative conditional, but makes that world epistemically impossible for Peter.

  8. The set theoretic version of the material conditional is Ac∪B.

  9. It is a general result that, if p indicates a credence function, p(A) ≥ p(A|\( \neg \)A\( \vee \)B) for any A and B. Equality occurs only if p(A) = 1 (Popper and Miller 1983).

  10. Strictly speaking, E cannot be as small as A∩B because Cr(\( \neg \)A) is not 0, but no harm is done by assuming that it can be that small.

  11. This implication does not exist in Stalnaker's first exposition of his theory (Stalnaker 1968), but becomes prominent in his specific application to indicative conditionals (Stalnaker 1975) where he introduces the requirement that f(P, w) is epistemically possible if w is. I will describe this implication in more detail in Sect. 3.1.

  12. Cr(B|A) ≈ 1 implies that Cr(¬B\( \wedge \)A) ≈ 0. Hence, Cr(¬(¬B\( \wedge \)A))  = Cr(\( \neg \)A\( \vee \)B) ≈ 1.

  13. See footnote 4.

  14. Water the aspidistras, according to Edgington (1995). In this case, A is “the butler did not do it”, and B “the gardener did it”.

  15. Neither, in fact does Adams', for Cr(\( \neg \)A\( \wedge \)B) = 1 − εCr(A) if Cr(B|A) = 1 − ε.

  16. I do not argue that all indicative conditionals should be interpreted that way. If someone tells me “if you drop that glass it will break”, I do interpret it as “if I drop this glass it will break and if I don't drop it it still would break if, counterfactually, I did drop it”.

  17. The case of Cr(¬B|A) being comparable to or less than ε is conceptually complex because it raises the question of what an agent should do when she is told a statement that, on Adams' account, she already agrees with. Furthermore, it is not clear how the agent should decide that she is in this position because the utterance of the statement does not tell her what ∈ is, other than that it is small in the opinion of the utterer. I will simply assume that Cr(\( \neg B|A \)) is not small by the agent's own light.

  18. This result leads, after inserting the expressions for u and v, to the same result as reported in (Huisman 2015).

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Acknowledgments

This article has benefitted greatly from several discussions with Stefan Lukits.

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    Leendert M. Huisman

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Huisman, L.M. Learning from Simple Indicative Conditionals. Erkenn 82, 583–601 (2017). https://doi.org/10.1007/s10670-016-9833-7

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