Posted: Dec 18, 2011 7:09 am
Thommo wrote:jlowder wrote:Hmmm... I may have oversimplified in an earlier post. In this example, I would say that the ratio of Pr(2 heads in a row | heliocentrism) to Pr(not 2 heads in a row | heliocentrism) is 1, which is just a fancy way of saying that heliocentrism is irrelevant.
Assuming that this is a typo (or not a typo, but a miswriting or whatever the technical word is) for
Pr(2 heads in a row | heliocentrism) to Pr(2 heads in a row | ¬heliocentrism) is 1
then I surely agree with that. My problem remains though, that according to this definition your class of "predictions" overlaps with your class of "irrelevancies" which to me seems like the definition of prediction doesn't capture what we would intend it to.
"H predicts E just in case H predicts that E is more likely than not, i.e., Pr(E|H&B) > 0.5."
It would seem more natural to define it in terms more like those you use at the end here, i.e. in terms of the ratio of the probability of the evidence given the hypothesis divided by the probability of the evidence given the falsity of the hypothesis, or informally that:
(1) E is evidence for H
is equivalent to
(2) H predicts E
is equivalent to
(3) Pr(E|H)/Pr(E) > 1
Although there may be the odd issue here with the non-finite sample space cases.
Edit: Removed a mis-pasted text fragment.
Hi Thommo -- Thanks for that well-written reply. You are correct; my original post contains the exact typo you identified. Further, I agree with the way you fixed it.
I see your point regarding the symmetry between (1), (2), and (3). I don't have much to say at this point, other than to say that at this time and in my "language," I am willing to allow E to be "evidence" for H even if H does not "predict" E (as I have defined those terms). I'm not locked into this position and may change it as I read more philosophy of science and confirmation theory. In fact, now that I think about it, I can think of at least one philosopher of science who argued at length that even Pr(E|H)>0.5 is too weak of a condition for E to count as evidence. So, on his scheme, my standards for "evidence" is unacceptably weak. I'm still digesting his arguments, so I consider my current terminology a work in progress and subject to change. But, to return to your comments, your suggestion is very reasonable and respectable.