I don’t like this analysis much; in particular “the odds of you being correct are pretty high; there’s really nothing unlikely in that scenario at all” seems unclear. Here’s what I consider a better one. (It’s similar to what fela says in a comment from last month but with more detail.)
There are multiple different mechanisms that would lead to you saying “Last night’s lottery numbers were [whatever]”. You might have just made them up at random. You might have tried to determine them by magic, prayer, ESP, etc., and not checked against reality. You might have done one of those things but actually overheard the numbers without noticing, and been influenced by that. You might have read a report of what they were. Etc.
It seems likely (though it might be hard to check) that most of the time when someone reports a set of lottery numbers from the past it’s because they looked up what they were. In that case they’re probably right, and (if wrong) probably close to right. So when someone tells me the numbers were 4, 2, 9, 7, 8, 3, there’s an excellent chance that those really were the numbers.
If they tell me the numbers were 1, 2, 3, 4, 5, 6, some quite different mechanisms become more probable—they might be making them up for fun, have been misled by a hoax, etc. In this scenario I’d still reckon a posterior probability well over 10^-6 for the numbers actually being 1,2,3,4,5,6, but probably not over 10^-1 until I’d got some more evidence.
Similarly, if the person were clutching a lottery ticket with the numbers 4, 2, 9, 7, 8, 3 then I would be less inclined to believe that those really were the numbers—again, because of the increased probability that the person in question is lying, is self-deceiving, etc. (Especially if they had something to gain from convincing me that they held a winning lottery ticket.)
Now, make it tomorrow’s lottery instead of tonight’s. What’s different? The most important difference is that the formerly most likely mechanism (they read the numbers in the newspaper or whatever) is no longer possible. So now we’re left with things like ESP, cheating, divine inspiration, etc., all of which are (in my judgement and probably yours) extremely unlikely to lead them to give numbers that correlate in any way with the real winning numbers. And also just-making-it-up, also unlikely to correlate with the real numbers.
OK. Now, finally, what about William Lane Craig’s argument and Chris’s assessment of it?
I think WLC is pretty much correct that the prior probability of the numbers being 4,2,9,7,8,3 is 10^-6, and that if someone tells you those were the numbers this is approximately enough evidence to bring that up to (say) 10^-1 or better. We can extend this further—suppose someone tells me what purport to be digits 999000..999999 of pi; I’d expect there to be a very good chance that they’re all correct. (And if you’re uncomfortable with probabilities over the digits of pi, which are after all necessarily whatever they are, we could make it “the result of 3000 coin-flips” or something.) So “extraordinary claims require extraordinary evidence”, if that’s taken to mean “very low-prior-probability claims require earth-shattering evidence”, is wrong. And it’s possible for “ordinary” (i.e., not earth-shattering) evidence to give a reasonable person a pretty big posterior probability for something whose prior probability is less than that of (say) “something at least kinda-sorta resembling Christianity is correct”.
Accordingly, I think Chris’s assessment is too simple.
As with the lottery example, we need to look at the possible ways in which the testimony that reaches us might come to be what it is. What makes me pretty comfortable about disbelieving (say) the alleged resurrection of Jesus despite the testimony in its favour isn’t the mere fact that the resurrection is awfully improbable prior to that testimony (though it is), it’s the fact that, of the mechanisms that would lead to the existence of such testimony, the ones that don’t involve an actual resurrection are much more probable than the ones that do. Nothing comparable to this is true in the case of the lottery example—because the testimony we’ve got in that case really is very low-probability if those aren’t the right lottery numbers. It’s “extraordinary” in the sense of “very low probability if the claim is false”, just not in the sense of “clearly the kind of thing that we never hear in the ordinary course of events”.
The maxim that “extraordinary claims require extraordinary evidence” is still OK, if it’s expanded as follows. To support a low-prior-probability claim C, you need evidence that’s sufficiently more probable on C than on not-C and, in particular, evidence that’s very improbable conditional on not-C. If C is a specification of the results of 3000 coin-flips, then someone’s report of those results can easily be such evidence—but, e.g., it wouldn’t be if what they report is that the results were perfectly alternating H and T even though the flips were carried out “fairly” by someone not intending to cheat. The improbability of C is simply a matter of its specificity, and if the evidence has the same specificity then (in many cases) that’s enough to make C quite probable a posteriori. But if C is, say, “Jesus was genuinely dead for at least a day and then alive again” then the fact that some people report this having happened isn’t particularly improbable conditional on not-C and that’s what makes it not “extraordinary” enough.
I don’t like this analysis much; in particular “the odds of you being correct are pretty high; there’s really nothing unlikely in that scenario at all” seems unclear. Here’s what I consider a better one. (It’s similar to what fela says in a comment from last month but with more detail.)
There are multiple different mechanisms that would lead to you saying “Last night’s lottery numbers were [whatever]”. You might have just made them up at random. You might have tried to determine them by magic, prayer, ESP, etc., and not checked against reality. You might have done one of those things but actually overheard the numbers without noticing, and been influenced by that. You might have read a report of what they were. Etc.
It seems likely (though it might be hard to check) that most of the time when someone reports a set of lottery numbers from the past it’s because they looked up what they were. In that case they’re probably right, and (if wrong) probably close to right. So when someone tells me the numbers were 4, 2, 9, 7, 8, 3, there’s an excellent chance that those really were the numbers.
If they tell me the numbers were 1, 2, 3, 4, 5, 6, some quite different mechanisms become more probable—they might be making them up for fun, have been misled by a hoax, etc. In this scenario I’d still reckon a posterior probability well over 10^-6 for the numbers actually being 1,2,3,4,5,6, but probably not over 10^-1 until I’d got some more evidence.
Similarly, if the person were clutching a lottery ticket with the numbers 4, 2, 9, 7, 8, 3 then I would be less inclined to believe that those really were the numbers—again, because of the increased probability that the person in question is lying, is self-deceiving, etc. (Especially if they had something to gain from convincing me that they held a winning lottery ticket.)
Now, make it tomorrow’s lottery instead of tonight’s. What’s different? The most important difference is that the formerly most likely mechanism (they read the numbers in the newspaper or whatever) is no longer possible. So now we’re left with things like ESP, cheating, divine inspiration, etc., all of which are (in my judgement and probably yours) extremely unlikely to lead them to give numbers that correlate in any way with the real winning numbers. And also just-making-it-up, also unlikely to correlate with the real numbers.
OK. Now, finally, what about William Lane Craig’s argument and Chris’s assessment of it?
I think WLC is pretty much correct that the prior probability of the numbers being 4,2,9,7,8,3 is 10^-6, and that if someone tells you those were the numbers this is approximately enough evidence to bring that up to (say) 10^-1 or better. We can extend this further—suppose someone tells me what purport to be digits 999000..999999 of pi; I’d expect there to be a very good chance that they’re all correct. (And if you’re uncomfortable with probabilities over the digits of pi, which are after all necessarily whatever they are, we could make it “the result of 3000 coin-flips” or something.) So “extraordinary claims require extraordinary evidence”, if that’s taken to mean “very low-prior-probability claims require earth-shattering evidence”, is wrong. And it’s possible for “ordinary” (i.e., not earth-shattering) evidence to give a reasonable person a pretty big posterior probability for something whose prior probability is less than that of (say) “something at least kinda-sorta resembling Christianity is correct”.
Accordingly, I think Chris’s assessment is too simple.
As with the lottery example, we need to look at the possible ways in which the testimony that reaches us might come to be what it is. What makes me pretty comfortable about disbelieving (say) the alleged resurrection of Jesus despite the testimony in its favour isn’t the mere fact that the resurrection is awfully improbable prior to that testimony (though it is), it’s the fact that, of the mechanisms that would lead to the existence of such testimony, the ones that don’t involve an actual resurrection are much more probable than the ones that do. Nothing comparable to this is true in the case of the lottery example—because the testimony we’ve got in that case really is very low-probability if those aren’t the right lottery numbers. It’s “extraordinary” in the sense of “very low probability if the claim is false”, just not in the sense of “clearly the kind of thing that we never hear in the ordinary course of events”.
The maxim that “extraordinary claims require extraordinary evidence” is still OK, if it’s expanded as follows. To support a low-prior-probability claim C, you need evidence that’s sufficiently more probable on C than on not-C and, in particular, evidence that’s very improbable conditional on not-C. If C is a specification of the results of 3000 coin-flips, then someone’s report of those results can easily be such evidence—but, e.g., it wouldn’t be if what they report is that the results were perfectly alternating H and T even though the flips were carried out “fairly” by someone not intending to cheat. The improbability of C is simply a matter of its specificity, and if the evidence has the same specificity then (in many cases) that’s enough to make C quite probable a posteriori. But if C is, say, “Jesus was genuinely dead for at least a day and then alive again” then the fact that some people report this having happened isn’t particularly improbable conditional on not-C and that’s what makes it not “extraordinary” enough.