I would like to share a doubt with you. Peter Achinstein, in his The Book of Evidence considers two probabilistic views about the conditions that must be satisfied in order for e to be evidence that h. The first one says that e is evidence that h when e increases the probability of h when added to some background information b:
(Increase in Probability) e is evidence that h iff P(h|e&b) > P(h|b).
The second one says that e is evidence that h when the probability of h conditional on e is higher than some threshold k:
(High Probability) e is evidence that h iff P(h|e) > k.
A plausible way of interpreting the second definition is by saying that k = 1/2. When one takes k to have such fixed value, it turns out that P(h|e) > k has the same truth-conditions as P(h|e) > P(~h|e) - at least if we are assuming that P is a function obeying Kolmogorov's axioms of the probability calculus. Now, Achinstein takes P(h|e) > k to be a necessary but insufficient condition for e to be evidence that h - while he claims that P(h|e&b) > P(h|b) is neither necessary nor sufficient for e to be evidence that h. That may seem shocking for those that take the condition fleshed out in (Increase in Probability) at least as a necessary condition for evidential support (I take it that the claim that it is necessary and sufficient is far from accepted - presumably one also wants to qualify e as true, or as known, or as justifiably believed, etc). So I would like to check one of Achinstein's counter-examples to the claim that increase in probability is a necessary condition for evidential support.
The relevant example is as follows:
The lottery counterexample
Suppose one has the following background b and piece of evidence e1:
b: This is a fair lottery in which one ticket drawn at random will win.
e1: The New York Times reports that Bill Clinton owns all but one of the 1000 lottery tickets sold in a lottery.
Further, one also learns e2:
e2: The Washington Post reports that Bill Clinton owns all but one of the 1000 lottery tickets sold in a lottery.
So, one has evidence in favor of
h: Bill Clinton will win the lottery.
The point now is that, although it seems right to regard e2 as being evidence in favor of h, it fails to increase h's probability conditional on (b&e1) - at least so says Achinstein. According to his example, the following is true:
P(h|b&e1&e2) = P(h|b&e1) = 999/1000.
Well, I have my doubts about this counterexample. The problem with it seems to me to be this: that e1 and e2 are taken to be the same piece of evidence. Let me explain. If e1 and e2 increase the probability of h, that is because they increase the probability of a further proposition:
g: Bill Clinton owns all but one of the 1000 lottery tickets sold in a lottery,
and, as it happens, g increases the probability of h. That The New York Times reports g, assuming that the New York Times is reliable, increases the probability of g - and the same can be said about The Washington Post reporting g. But the counterexample seems to assume that both e1 and e2 are equivalent with g, and they're not. Now, it is clear that P(h|b&g) = P(h|b&g&g), but this does not show that e2 fails to increase h's probability on (b&e1). So, if it is true that e2 increases the probability of g conditional on e1, that is, if P(g|e1&e2) > P(g|e1), and if it is true that g increases the probability of h, then it is also true that e2 increases the probability of h. I may be missing something, but this reasoning sounds right to me - the example wouldn't be a counterexample. What do you think?
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I think your example eviscerates the first "Increase in Probability" definition, at least as presented here, and shows that it doesn't account for non-independent evidence. If I wake up one morning and read in the New York Times reports that Bill Clinton has bought 99.9 % of tickets in a lottery, this strongly increases my estimate of the probability that Clinton will win the lottery. Reading the same story in the Washington Post does not materially increase my estimate given the background information in the New York Times. (I suppose it slightly reduces the probability that the Times story is a hoax or mistaken. Just maybe that's relevant.) Thus by this definition the Post story is not evidence (or at least is very weak evidence) that Bill Clinton will win the lottery.
However, suppose instead I wake up and read the Post story first. This now provides strong evidence that Bill Clinton will the lottery. The Times story is weak evidence at best. So depending on the irrelevant detail of which story I read first, one is strong evidence and one is weak evidence? That seems wrong. I don't want the strength of evidence to depend more on the irrelevant detail of which order I encounter two pieces of evidence. So perhaps what's being defined here is not the quality of the evidence but the usefulness of new evidence to me given what I already know?
Of course evidence and probabilities, especially non-independent probabilities are not additive.
This is not inobvious, so I notice that I am confused. I have to think that I'm misunderstanding this definition; or that there are details in the book that you're not reporting.
So, I'll kind of second the observation in the comment above. It seems to me that, from the fact that reading the same story in the Washington Post does not make your epistemic situation better, it does not seem to follow that the Post story is not evidence that Bill win the lottery. That is: from the fact that a certain piece of evidence is swamped by another piece of evidence in a certain situation, it does not follow that the former is not evidence. We can see that it is evidence just following your steps: we conceive another situation where I didn't read the Times story but I read the Post story - and it is evidence that Bill win the lottery in this situation.
I agree that it seems just wrong to grant that strong evidence and weak evidence is determined by the access we have to evidence in order of time. But from the fact that one does not gain more justification to believe h by learning e it does not follow that e is evidence that h, all things considered.