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.

I concur. Now consider the case where you observed the ticket sales, and you saw Bill buy 999 tickets (g). Then someone tells you that Bill bought 999 tickets(e1). Is e1 evidence that Bill will win the lottery?

Suppose that you saw the ticket sales, and saw someone other than Bill buy 501 out of 1000 tickets. Is there any possible evidence that Bill will win? (Assume tickets are nontransferable & other loopholes are accounted for-the odds are 501:499 against)

Suppose that you didn't see the ticket sales, but a reliable source (The New York Times) reports that Bill bought somewhere around 500 tickets, but they don't know the exact number. Would that be evidence that Bill will win? (Assume that there are many people as eligible to buy tickets)

I will continue using the definition "e is evidence of h iff P(h|e) > P(h)". I don't think that P(h|b) is meaningful unless P(h|~b) is also meaningful.

Have you read this?

Your interpretation about g is correct.

The high probability interpretation is not a useful interpretation of "evidence," and there's a much easier way to discuss why: implication. P("A or ~A"|"My socks are white")=1, because P("A or ~A") is 1, and conditioning on my socks being white cannot make that less true. It is not sensible to describe the color of my socks as evidence for the truth value of "A or ~A".

The increase in probability definition is sensible, and what is used locally for Bayesian evidence.

While I find this particular re-definition a bit silly, that doesn't mean that in general having more succinct definitions isn't a good thing.

If the second definition of evidence were used, it would mean that "collecting evidence" would be a fundamentally different thing than it would be in the first case. In the second, "evidence" is completely relative to what is already known, and lots of new material would not be included.

So if I go out and ask people to "collect evidence", their actions should be different depending on which definition we collectively used.

In addition, the definition would lead to interesting differences in quantities. If we used the first definition, me having "lots of evidence" could mean having lots of redundant evidence for one small part of something (of course, it would also be helpful to quantify "lots", but I believe that or something similar could be done). In the second definition, I imagine it would be much more useful to what I actually want.

This new definition makes "evidence" much more coupled to resulting probabilities, which in itself could be a good thing. However, it seems like an unintuitive stretch for my current understanding of the word, so I would prefer that rather than re-defining the word, a condition were used. For example, "updating evidence" for the second definition.

What do you think?

I think philosophers need to spend less time trying to come up with necessary-and-sufficient definitions of english words.

So, I would like to thank you guys for the hints and critical comments here - you are helping me a lot! I'll read what you recommended in order to investigate the epistemological properties of the degree-of-belief version of bayesianism. For now, I'm just full of doubts: "does bayesianism really stand as a normative theory of rational doxastic attitudes?"; "what is the relation between degrees of belief and evidential support?", "is it correct to say that people reason in accordance to probability principles when they reason correctly?", "is the idea of defeating evidence an ilusion?", and still others. =]

Well, you didn't answered to the puzzle.

So, in order to answer the puzzles, you have to start with probabilistic beliefs, rather than with binary true-false beliefs. The problem is currently somewhat like the question "is it true or false that the sun will rise tomorrow." To a very good approximation, the sun will rise tomorrow. But the earth's rotation could stop, or the sun could get eaten by a black hole, or several other possibilities that mean that it is not absolutely known that the sun will rise tomorrow. So how can we express our confidence that the sun will rise tomorrow? As a probability - a big one, like 0.999999999999.

Why not just round up to one? Because although the gap between 0.999999999999 and 1 may seem small, it actually takes an infinite amount of evidence to bridge that gap. You may know this as the problem of induction.

So anyhow, let's take problem 1. How confident are you in P1, P2, and P3? Let's say about 0.99 each - you could make a hundred such statements and only get one wrong, or so you think. So how about T? Well, if it follows form P1, P2 and P3, then you believe it with degree about 0.97.

Now Ms. Math comes and tells you you're wrong. What happens? You apply Bayes' theorem. When something is wrong, Ms. Math can spot it 90% of the time, and when it's right, she only thinks it's wrong 0.01% of the time. So Bayes' rule says to multiply your probability of ~T by 0.9/(0.030.9 + 0.970.0001), giving an end result of T being true with probability only about 0.005.

Note that at no point did any beliefs "defeat" other ones. You just multiplied them together. If Ms. Math had talked to you first, and then you had gotten your answer after, the end result would be the same. The second problem is slightly trickier because not only do you have to apply probability theory correctly, you have to avoid applying it incorrectly. Basically, you have to be good at remembering to use conditional probabilities when applying (AME).

I can conceive the puzzle as one where all the relevant beliefs - (R1), (T), (AME), etc, - have degree 1.

I suspect that you only conceive that you can conceive of that. In addition to the post linked above, I would suggest reading this, and this, and perhaps a textbook on probability. It's not enough for something to be a belief for it to be a probability - it has to behave according to certain rules.

"T is true; therefore, evidence that it is false is false. This constitutes invalid reasoning, because it rules out new knowledge that may in fact render it truly false."

Actually, I think if "I know T is true" means you assign probability 1 to T being true, and if you ever were justified in doing that, then you are justified in assigning probability 1 that the evidence is misleading and not even worth to take into account. The problem is, for all we know, one is never justified in assigning probability 1 to any belief. So I'd say the problem is a wrong question.

Edited: I meant probability 1 of misleading evidence, not 0.

I can conceive the puzzle as one where all the relevant beliefs - (R1), (T), (AME), etc, - have degree 1.

Well, in that case, learning RM & TM leaves these degrees of belief unchanged, as an agent who updates via conditionalization cannot change a degree of belief that is 0 or 1. That's just an agent with an unfortunate prior that doesn't allow him to learn.

More generally, I think you might be missing the point of the replies you're getting. Most of them are not-very-detailed hints that you get no such puzzles once you discard traditional epistemological notions such as knowledge, belief, justification, defeaters, etc. (or change the subject from them) and adopt Bayesianism (here, probabilism & conditionalization & algorithmic priors). I am confident this is largely true, at least for your sorts of puzzles. If you want to stick to traditional epistemology, a reasonable-seeming reply to puzzle 2 (more within the traditional epistemology framework) is here: http://www.philosophyetc.net/2011/10/kripke-harman-dogmatism-paradox.html