Originally Posted by Wayfaerer
That's all fine but I can still see how evidence for one of these two completely different claims does not support the other.
Well... it's clearly not all fine if you still think that you can imagine such evidence. It would be like saying that you can imagine a situation where a person is an unmarried man but is not a bachelor. Obviously there can be no such situation simply due to our definitions of "unmarried man" and "bachelor." For the present case, could you illustrate for us how some event could be evidence for T but not S, or vice versa?
Originally Posted by Wayfaerer
How can you rigorously prove this? It seems to me you just disproved this notion. Why not conclude this kind of thinking is wrong in the first place rather than seeing it as some confounding logical glitch?
The proof (summarized from Carl Hempel) was laid out pretty straightforwardly in my first posthowever, I don't want to defend that proof too much because I do actually think that this paradox can be resolved in a satisfying way by taking a Bayesian perspective on the problem, a perspective which I think is actually quite intuitive. It would take a fair amount of space to fully explicate the Bayesian argument (it is based on Bayes' Theorem, a result from probability theory), but I think the important points can be summed up adequately.
Recall that the Tacking Paradox depends on two assumptions about how evidence can bear on theory: (1) if T entails E, then E is evidence for T; and (2) if E is evidence for T, and T entails P, then E is evidence for P. If we accept both of these seemingly innocent assumptions, then the paradox can always be derived.
Basically, the Bayesian accepts (1) but does not fully accept (2). Instead, the Bayesian accepts that (2) is sometimes true, but denies that it must always be the case: if E is evidence for T & P, then E is allowed to provide different degrees of support for T and for P separately, says the Bayesian. Crucially, E serves as evidence for T or P only to the extent that E would be correspondingly less likely to have obtained if T or P were NOT true. With respect to P, for example, the Bayesian says that if E is equally likely to have occurred whether it is the case that T & P or that T & notP, then E cannot be considered as evidence for or against P, whether or not it may be evidence for T. In this way the Tacking Paradox is blocked.
To make this more concrete let's return to my running example of the roommate. We had arrived at the paradoxical conclusion that, because my observation that my roommate had slept through his alarm served as evidence for the joint hypothesis that "my roommate was out drinking last night AND the moon is made of green cheese" (by [1] from above), then it must also be the case that my roommate having slept through his alarm must also be evidence for the simple hypothesis that "the moon is made of green cheese" (by [2] from above). However, the Bayesian points out that the probability that I would observe my roommate to sleep through his alarm given that "my roommate was out drinking last night AND the moon is made of green cheese" is exactly the same as the probability that I would observe my roommate to sleep through his alarm given that "my roommate was out drinking last night AND the moon is NOT made of green cheese." This is of course not true for the first part of the joint hypothesis, the part saying that my roommate was out drinking last night. In that case, the evidence is undeniably more likely to have obtained if my roommate had been out drinking last night then if he had not. So we can say that while my observation of my roommate sleeping through his alarm DOES serve as evidence that he was out drinking last night, it does NOT serve as evidence that the moon is made of green cheese.
I think you will agree that this is a pretty satisfying answer to the paradox.
The Bayesian also has a response to the Raven's Paradox, but it is somewhat less satisfying and also more difficult to explain without appealing directly to Bayes' Theorem, which I want to avoid. Roughly put, the Bayesian bites the bullet and concedes that observing any nonblack thing that is a nonraven does indeed serve as some degree of evidence that all ravens are black. However, the Bayesian maintains that, for any realistic figures, such an observation provides such an infinitesimally small degree of support for the hypothesis that we can say that it has no practical bearing on the hypothesis "in the real world." So the Raven's Paradox is not blocked in the same way that we managed to block the Tacking Paradox, but its impact is lessened considerably.


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