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A weaker condition for transitivity in probabilistic support

  • Original paper in Philosophy of Probability
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Abstract

Probabilistic support is not transitive. There are cases in which x probabilistically supports y, i.e., Pr(y | x) > Pr(y), y, in turn, probabilistically supports z, and yet it is not the case that x probabilistically supports z. Tomoji Shogenji, though, establishes a condition for transitivity in probabilistic support, that is, a condition such that, for any x, y, and z, if Pr(y | x) > Pr(y), Pr(z | y) > Pr(z), and the condition in question is satisfied, then Pr(z | x) > Pr(z). I argue for a second and weaker condition for transitivity in probabilistic support. This condition, or the principle involving it, makes it easier (than does the condition Shogenji provides) to establish claims of probabilistic support, and has the potential to play an important role in at least some areas of philosophy.

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Notes

  1. See, e.g., Eells and Sober (1983, pp. 43–44), Hanen (1971), Hesse (1970, pp. 50–51), and Shogenji (2003, p. 613).

  2. This reads: Pr(r | h) = 1, 1 > Pr(r), and Pr(r) = 1/2.

  3. Better put: Is there a nontrivial such condition? Clearly, there is a trivial such condition, e.g., the condition that Pr(z | x) > Pr(z).

  4. All references to Shogenji are to Shogenji (2003).

  5. A similar principle, though about probabilistic causality, not probabilistic support, is established by Eells and Sober (1983). Also, see Reichenbach (1956, Ch. IV, sec. 19) and Sober (2009). For discussion of (TPS) as it relates to the transmission of confirmation by coherence, see Dietrich and Moretti (2005).

  6. Further, since Pr(h & ~r) = 0, and since, thus, Pr(d | h & ~r) is undefined, it follows that it is not the case that Pr(d | h & ~r) = Pr(d | ~r).

  7. Shogenji considers a more controversial case of the sort in question, where the first claim says that there is testimony that a certain miracle occurred, the second claim says that the miracle in question occurred, and the third claim says that God exists. Shogenji argues that, in such a case, (C) holds and so, supposing that the first claim probabilistically supports the second, and that the second claim probabilistically supports the third, it follows, by (TPS), that the first claim probabilistically supports the third. Cf., e.g., Holder (1998), Otte (1993), and Schlesinger (1987).

  8. Mary Hesse (1970, pp. 54–55) establishes a similar principle. It can be put as follows: If (1) Pr(y | x) > α, (2) Pr(z | y) > β, and (3) Pr(z | x & y) ≥ Pr(z | y), then Pr(z | x) > αβ. The antecedent of this principle, like the antecedent of (TPS*), does not require that Pr(z | x & y) = Pr(z | y), and does not require that Pr(z | x & ~y) = Pr(z | ~y). But note: It is not the case that when the antecedent of Hesse’s principle is satisfied, and when α = Pr(y) and β = Pr(z), it follows that Pr(z | x) > Pr(z). Suppose Smith selects a card at random from a standard deck of cards. Let “d,” “h,” and “r” be understood as in Case 1. Then, Pr(h | r) > Pr(h), Pr(~d | h) > Pr(~d), and Pr(~d | r & h) ≥ Pr(~d | h). Thus, by Hesse’s principle, Pr(~d | r) > Pr(h)Pr(~d). But, Pr(~d | r) < Pr(~d).

  9. The labeling below, “(a),” “(b),” etc., is mine, not Shogenji’s.

  10. The 8-ball is neither low-numbered nor high-numbered. The high-numbered balls are: 9-ball, 10-ball, . . .,15-ball.

  11. I am assuming that Pr(t & ~i) > 0.

  12. I thank Shogenji (private communication) for suggesting to me a case of this sort. Some of the cases of “useful false beliefs” given in Klein (2008), e.g., the case of “Mr Butterfingers” (p. 51), can be modified so as to have all the relevant features of Case 5.

  13. For discussion of the distinction between evidential support in the relevance sense versus evidential support in the absolute sense, and of the related distinction between having evidence in the relevance sense versus having evidence in the absolute sense, see Okasha (1999).

  14. I do not mean for this to be an adequate formalization of the issue Crispin Wright (2002, 2003) has in mind in speaking of when it is that warrant transmits across entailment. For discussion of how to formalize the issue Wright has in mind, see Chandler (2010), Moretti (2010), and Okasha (2004). Cf. Pynn (2011).

  15. See, e.g., Mackie (1969, p. 36), Okasha (1999, p. 45), and White (2006, sec. 5). Note, however, that there are no cases in which Pr(y | x) is high, y entails z, and yet Pr(z | x) is not high. See Okasha (1999) and Salmon (1965).

  16. Dretske (1970) can be read along these lines; see Okasha (1999).

  17. See White (2006, sec. 5) for defense of this sort of point.

  18. I am assuming, as seems plausible, that justification is truth-conducive at least in the sense that justification implies an increase in the probability of truth. The question of whether coherentist justification implies an increase in the probability of truth is to be distinguished from the question of whether coherentist justification implies a high probability of truth, and from the question of whether, ceteris paribus, greater coherence implies a greater probability of truth.

  19. What information is to be included in the background information codified in Pr is a difficult issue. I discuss it elsewhere (2010, 2012).

  20. This case improves on the case I had in its place in a prior version of this paper. Thanks to an anonymous reviewer for help here.

  21. A different sort of case is where (C*) holds, and Pr(z | x) > Pr(z), but Pr(y | x) < Pr(y) and Pr(z | y) < Pr(z). Suppose a ball is randomly selected from a standard set of billiard balls, “w” and “e” are understood as in Case 3, and “n” is the claim that the ball selected is an even-numbered ball. It follows that Pr(e | w & n) Pr(e | n), Pr(e | w & ~n) Pr(e | ~n), and Pr(e | w) > Pr(e), but Pr(n | w) < Pr(n) and Pr(e | n) < Pr(e). Cases of the sort in question are interesting but do nothing to undermine the point that (TPS*) is correct and makes it easier than does (TPS) to establish claims of probabilistic support.

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Acknowledgments

I wish to thank Nicholaos Jones, Tomoji Shogenji, Joshua Smith, and an anonymous reviewer for very helpful comments on prior versions of this paper.

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Roche, W.A. A weaker condition for transitivity in probabilistic support. Euro Jnl Phil Sci 2, 111–118 (2012). https://doi.org/10.1007/s13194-011-0033-7

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