Posted: Dec 18, 2011 7:56 am
jlowder wrote: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.
Indeed, that would be of concern, but as I suggested, I think the fact that it can classify irrelevant hypotheses, such as in the heliocentrism/coin toss example, as "evidenced by E" (or even worse hypotheses where updating the prior probability with the evidence actually makes the hypothesis less likely!) is more critical. The exact threshold of increase can be easily amended in the suggestion* by inserting a constant in the equation:-
(3) Pr(E|H)/Pr(E) > c, c∈ℝ, c≥1
or by amending it to comparing the proportional increase in probability of that which could be made by the ideal evidence and that which has been made, i.e.
(3) (Pr(E|H)-Pr(E))/(1-Pr(E)) > c (0 < c < 1)
But I'm probably just repeating myself now. Thanks for your replies and clarifications!
*as taken as an off the top of ones head example - I'm not especially convinced of it myself, especially outside of the finite case - it's worth mentioning there is some obvious potential for fun with "divide by 0" errors when we go countable and beyond, as in the general case P(x) = 0 is not the same thing as "x is impossible".
ETA: One final clarification, my earlier comments all apply only to theory, obviously in reality it's necessary for a prediction to chronologically precede it's effect, something I jovially overlooked here as unimportant, but human bias and the need to obey the laws of causality can affect what can be practically considered a prediction once we move off of the "chalk board".