I don't understand what it would mean to divorce a hypothesis h from the background b.

Suppose you have the flu (background b); there is zero chance that you don't have the flu, so P(~b)=0 and P(x&~b)=0, therefore P(x|~b)=0 (or undefined, but can be treated as zero for these purposes).

Since P(x)=P(x|b)+P(x|~b), P(x)=P(x|b) EDIT: As pointed out below, P(x)=P(x|b)P(b)+P(x|~b)P(~b). This changes nothing else . If we change the background information, we change b and are dealing with a new hypothetical universe (for example, one in which taking both Fluminex and Fluminalva increases the duration of a flu.)

In that universe, you need prior beliefs about whether you are taking Fluminex and Fluminalva, (and both, if they aren't independent) as well as their effectiveness separately and together, in order to come to a conclusion.

P, h, and e are all dependent on the universe b existing, and a different universe (even one that only varies in a tiny bit of information) means a different h, even if the same words are used to describe it. Evidence exists only in the (possibly hypothetical) universe that it actually exists in.

I think that we agree that neither of the definitions offered in the post are correct.

Can you see any problem with "e is evidence of h iff P(h|e) > P(h)", other than cases where evidence interacts in some complex manner such that P(h|e1)>P(h); P(h|e2)>P(h); but P(h|e1&e1)<P(h) (I'm not sure that is even possible, but I think it can be done with three mutually exclusive hypotheses).

Well, duh. You're right, the post was pretty clear about this. I need to read more carefully. So does he believe that the second condition is both necessary and sufficient? That seems prone to a bunch of counterexamples also.

It's not clear to me why exactly you want the definition of evidence to not rely on the particular background of the mind where the P resides.

If you limit b to tautologies, you kill its usefulness. "This is a fair lottery in which one ticket drawn at random will win" isn't a tautology.

My first case is where g is given- you know it as well as you know anything else, including the other givens. I would not say that whether a particular fact is evidence depends on the order in which you consider them.

Do you concur that "It is 75% likely that Bill will win the lottery given 'it is not the case that This is a fair lottery in which one ticket drawn at random will win. . . .'" p(h|~b) is not a meaningful statement?

This is not a case where we have two definitions talking about two sorts of things (like sound waves versus perception of sound waves). This is a case where we have two rival mathematical definitions to account for the relation of evidential support.

What is this "relation of evidential support", that is a given thing in front of us? From your paraphrase of Achinstein, and the blurb of his book, it is clear that there is no such thing, any more than "sound" means something distinct from either "vibrations" or "aural perceptions". "Sound" is a word that covers both of these, and since both are generally present when we ordinarily talk of sound, the unheard falling tree appears paradoxical, leading us to grasp around for something else that "sound" must mean. "Evidence" is a word that covers both of the two definitions offered, and several others, but the fact that our use of the word does not seem to match any one of them does not mean that there must be something else in the world that is the true meaning of "evidence".

The analogy with unheard falling trees is exact.

What would you expect to accomplish by discovering whether some particular e really is "evidence" for some h, that would not be accomplished by discovering whether each of the concrete definitions is satisfied? If you know whether e is "fortitudinence" for h (increases its probability), and you know whether e is "veritescence" for h (gives a posterior probability above 1/2), what else do you want to know?

BTW, around here "fortitudinence" is generally called "Bayesian evidence" for reasons connected with Bayes theorem, but again, that's just a definition. There are reasons why that is an especially useful concept, but however strong those reasons, one is not discovering what the word "evidence" "really means".

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.

There's a deep philosophical point at stake here. Is probability a 1) quantification of a person's uncertainty, or 2) a statement about the universe?

2) is a position that is not well-regarded here, and I would recommend Probability is in the Mind and then possibly Probability is Subjectively Objective as to why.

If 1), then their previous knowledge matters. It's one particular person's uncertainty. You start off very uncertain about the lottery result. Then you read the first newspaper report, which makes you much less uncertain about the lottery result. Then you read the second newspaper report, which makes you very slightly less uncertain about the lottery result. As you watch your uncertainty decrease, you see that the first report has a huge effect (and thus is strong evidence), and the second report has a small effect (and thus is weak evidence). With more background knowledge about correctness, the second report drops to 0 effect (and thus is not evidence to you).

The mathematical way to think about this is that the strength of evidence is a function of both e and b. This is necessary to ensure that there's a probability shift, and if you don't force a shift, you can have even more silly 'evidence', like sock color being evidence for logical tautologies.

One might object that they're not particularly interested in measuring their personal uncertainty, but in affecting the beliefs of others. If you wanted to convince someone else that Bill Clinton is probably going to win the lottery, it seems reasonable to be indifferent to whether they read the Times or the Post, so long as they read one. But your personal measure of evidence is wildly different between the two papers! How do we reconcile your personal measure of uncertainty, and your desire to communicate effectively?

The answer I would give is being more explicit about the background b. It's part of our function, and so let's acknowledge it. When b is "b & e1", then e2 is not significant evidence. When b is just b, e2 is significant evidence, and so if you want to convince someone else of h and their current knowledge is just b, you can be indifferent between e1 and e2 because P(h|b,e1)=P(h|b,e2).

Let's further demonstrate that with a modification to the counterexample like the one Manfred suggested. Suppose I learn e1, that the New York Times reports that Bill Clinton owns all but one of the tickets, by reading that day's copy of the Times, which I'll call r1.

Suppose my background, which I'll call t, is that the New York Times (and my perception of it) is truthful. So P(e1|t,r1)=1. I liked the prose so much, I give the newspaper a second read, which I'll call r2. Is r2 evidence for e1? Well, P(e1|t,r1,r2)=1, which is the same as P(e1|t,r1), which suggests by the "increase in probability" definition that reading the newspaper a second time is not evidence for what the newspaper says. (Note that if we relax the assumption that my reading comprehension is perfect, then reading something a second time is evidence for what I thought it said, assuming I think the same thing after the second read. If we only relax the assumption that the newspapers are perfectly correct, we don't get a change in evidence.)

Does it seem reasonable that, given perfect reading comprehension, that considering the same piece of evidence twice should only move your uncertainty once? If so, what is the difference between that and the counterexample where one newspaper says "Times" on the front, and the other says "Post"?

(If you relax the assumption that the newspapers are perfectly correct, then the second newspaper is evidence by the "increase in probability" definition, because of the proposition g discussed in the OP.)

Well, it's a complete answer to the conundrum.

It seems very strange to me to look at e1 and e2 as point values without weights or confidence distributions. Taking no other information about e1 and e2 other than the fact that they both indicate a 999/1000 victory is quite limiting and makes for impossible meta-analysis. Other information could include what you think of both sources.

If you could get an accurate 90% confidence interval for each (or any accurate probability distribution) this could make a lot more sense. This must encompass the expected error in the New York Time's error margin, especially if they don't have one. For example, you may find that whenever reference a statistic, 90% of the time it is has at most 15% error compared to the true result (this would be really useful btw, someone should do this). Even if you estimate this number, you could still get a workable value.

If their 90% confidence interval was 0% error, and their reported statistics were always exactly true, then I do not believe you would update at all from e2.

I feel like it is possible to combine two 90% confidence intervals, and my guess is that any two with the same mean would result in higher certainty than at least the worst estimate (the one with the wider 90% confidence interval), possible higher than both. Solving this mathematically is something I'm not too sure about.