In my previous post, I suggested that we might distinguish between two “evidential favoring” relations. One is a “competitive” notion appropriate only for mutually exclusive hypotheses which the Law of Likelihood explicates. The other is a “comparative” notion that is appropriate for any pair of hypotheses which is explicated by a measure of confirmation together with the following “Bridge Principle:”
(†) Evidence E favors hypothesis H1 over hypothesis H2 if and only if it confirms H1 more than H2.
I now prefer a cleaner approach. To my ear, the phrase “favors H1 over H2” sounds appropriate only when H1 and H2 are competitors in the strong sense that only one of them can be correct. Not so with the phrase “confirms H1 more than H2.” Thus, my ear suggests rejecting (†) for hypotheses that are mutually exclusive.
I conjecture that wherever there seems to be a need for a notion of evidential favoring for hypotheses that are not mutually exclusive, the notion “confirms more than” can do the necessary work.
Take, for instance, the conjunction fallacy. Kahneman and Tversky told test subjects that a woman named Linda majored in philosophy and participated as a student in political demonstrations. They then asked the test subjects whether it was more probable that (a) Linda is a bank teller or (b) Linda is a bank teller and is active in the feminist movement. Most test subjects chose (b), which demonstrates fallacious reasoning because the probability that Linda is BOTH a bank teller AND active in the feminist movement cannot be higher than the probability that Linda is a bank teller, which does not preclude the possibility that she is also active in the feminist movement.
One possible explanation for this phenomenon is that test subjects were responding to the fact that the facts reported about Linda evidentially favor (b) over (a). This explanation would no longer be available if we restricted the use of the notion of evidential favoring to mutually exclusive hypotheses. However, this fact is not a good objection to that restriction because one could simply say instead that the facts reported about Linda confirm (b) more than they confirm (a), which seems to me a clearer way to state the point anyway.
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