Abstract
There are many scientific and everyday cases where (a) each of Pr(H 1 | E) and Pr(H 2 | H 1) is high and (b) it seems that Pr(H 2 | E) is high. But high probability (or absolute confirmation) is not transitive and so it might be in such cases that (a) each of Pr(H 1 | E) and Pr(H 2 | H 1) is high and (c) in fact Pr(H 2 | E) is not high. There is no issue in the special case where the following condition, which I call “C1”, holds: H 1 entails H 2. This condition is sufficient for transitivity in high probability. But many of the scientific and everyday cases referred to above are cases where it is not the case that H 1 entails H 2. I consider whether there are additional (non-trivial) conditions sufficient for transitivity in high probability. I consider three candidate conditions. I call them “C2”, “C3”, and “C2&3”. I argue that C2&3, but neither C2 nor C3, is sufficient for transitivity in high probability. I then set out some further results and relate the discussion to the Bayesian requirement of coherence.
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Notes
See, for example, Ellis (2008, p. 2.34).
Krauss seems to hold that Pr(M | I) is high but less than 1.
An example of a trivial condition sufficient for transitivity in high probability is this: Pr(H 2 | E) > t.
Nothing of importance in this section hinges on the choice of \( \sqrt[2]{t} \) as the threshold for super-high probability. THPC2 below is incorrect given any value for t and any alternative threshold for super-high probability (greater than t and less than 1). Things are different in Section 4. If the threshold for super-high probability were greater than \( \sqrt[2]{t} \), then THPC2 & 3 would still hold without exception but its antecedent would be stronger than it needs to be (in order for it to hold without exception). If the threshold for super-high probability were less than \( \sqrt[2]{t} \), then THPC2 & 3 would fail to hold without exception.
Consider the conditions:
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Condition 5 (C5):
Pr(H 2 | H 1 & E) = Pr(H 2 | H 1).
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Condition 6 (C6):
Pr(H 2 | H 1 & E) ≥ Pr(H 2 | H 1) and Pr(H 2 | ¬H 1 & E) ≥ Pr(H 2 | ¬H 1).
C4 is stronger than each of C5 and C6. Hence neither C5 nor C6 is sufficient for transitivity in high probability. Hence none of C3, C4, C5, and C6 is sufficient for transitivity in high probability. The situation is a bit different in the context of increase in probability: each of C4 and C6 but neither C3 nor C5 is sufficient for transitivity in increase in probability. See Atkinson and Peijnenburg (2013), Roche (2012a, b, 2014, 2015), Roche and Shogenji (2014), Shogenji (2003, forthcoming), and Sober (2015, Ch. 5) for relevant discussion.
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Condition 5 (C5):
That THP*C3 holds without exception can be seen by appeal to (7) above.
There is also a requirement to the effect that upon the receipt of new information a subject’s degree of belief function should be updated by conditionalization (strict conditionalization, Jeffrey conditionalization, or Field conditionalization). For discussion of Bayesianism, and for references, see Talbott (2016).
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Acknowledgments
I wish to thank two anonymous reviewers, Tomoji Shogenji, and an audience at the 54th Annual Meeting of the Alabama Philosophical Society for helpful comments on prior versions of the paper.
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Roche, W. A condition for transitivity in high probability. Euro Jnl Phil Sci 7, 435–444 (2017). https://doi.org/10.1007/s13194-017-0172-6
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DOI: https://doi.org/10.1007/s13194-017-0172-6