Putting Logic in its Place: Formal Constraints on Rational Belief by David Christensen

Author:David Christensen [Christensen, David]
Language: eng
Format: epub
ISBN: 0199263256

14 This assumes that we have no reason to trust some elements more than others, and that only one of the elements is false. If the events related by a particular judgment are the ends of a 7-link chain, then there's a 7/26 chance that the error is in one of those links (in which case the judgment is false); thus, there is only a 19/26 chance that it's true.

end p.90

Now I think that it is not at all clear that P6 is unworthy of belief. But before thinking more carefully about this question, I'd like to fill the case out in a bit more detail, the better to fix our intuitions. First, the circular structure of the elements may be doing some intuitive work by suggesting that there is at least one big mistake in the elements. If that's right, we may well be influenced by the thought that the source from which we obtained evidence for the elements was not good, and that we shouldn't really be very confident of any of them. True, giving the elements 0.96 probability is mathematically possible in a case of this abstract structure. But that doesn't make 0.96 an intuitively realistic estimate of the probabilities in an actual case meeting the description. Let us, then, specify how the elements are arrived at. One might naturally imagine that the various events could somehow have been timed by relations to external events (e.g., the car was filled with gas at 8:15 am). But this would not lead to a circle of priority claims. To fix our intuitions as clearly as possible, let us try to fill in the abstract description in a fairly natural way, so that our evidence will lead to a circular structure with the high probabilities the argument requires. Here's one way of doing so (I've also taken the liberty of changing subject matter, to remove any distraction that might be caused if our supposition that we're reconstructing a serious accident for some important purpose had the effect of raising the intuitive standard for rational belief above 0.96).

Suppose there is a 26-person race, which we haven't seen. The rules stipulate that each racer will tell us who finished right behind her (and will tell us nothing else). The rules (which we may suppose are followed religiously) also stipulate that all racers tell us the truth, with the exception of the last finisher, who is to tell us that the racer who actually won finished behind her. We thus arrive at intuitively reasonable probabilities of 0.96+ for the elemental claims such as "racer C finished before racer D." Here, it is even less clear that belief in these claims would be irrational. Nevertheless, I think it must be acknowledged that many would be hesitant to assert unqualifiedly that D finished before E. And I think that some

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