Um, sorry, but seriously?! Arguing about definitions of words? This is entirely ridiculous and way below the minimum rationality that should be expected from posts on Less Wrong. Downvoted for proposing serious discussion of a topic that deserves no such thing. Since you seem sincere I'll try and give you a quick overview of the problems here, but you really need to reread the sequence "A Human's Guide to Words" to get a full picture.
First, while I have an answer to what the useful definition of evidence is (in the sense that it describes a useful feature of reality), I will refrain from pointing it out here because it is irrelevant to the topic at hand. If someone really needed the word "evidence" for some reason, including potential hypothesis-favoring-data sufficient to convince me that most people mean something very different from me by the word "evidence", I'd be willing to give up the word. After all, words don't have meanings, they're just mental paintbrush handles for someone else's brain, and if that handle paints the wrong picture then I'll use a different one.
That said, the thrust of the problem with your post is exactly the same as the definitional dispute over a tree in a forest. There is no "true" meaning of evidence, and anyone arguing about one is doing so with an intent to sneak in connotations to win an argument by appealing to basic human fallacies. Definitional Disputes are an indisputably Dark Side tactic; the person doing might be honest, but if so then they are severely confused. Most people couldn't identify the difference between good and bad epistemology if it hit them in the face, and this does not make them evil, but it does make them wrong. Why would anyone care what the "true meaning" of evidence is, when they could just break down the various definitions and use each consistently and clearly? The only reason to care runs along the lines of "evidence is an important concept [hidden inference], this is the true definition, therefore this definition is important", replacing "important" with something specific to some discussion.
Only think about words as paintbrush handles, and the problem goes away. You can then start focusing on the concept behind your handle, and trying to communicate instead of win. Once you and your audience can all understand what is being said - that is, when the pictures you draw in their brain match the pictures in your head - then you're done. If you dispute anything at that point, it will be your true dispute, and it will have a resolution - either you have different priors, one or more of you is irrational, or you will walk away in agreement (or you'll run out of time - humans aren't ideal reasoners after all). Play Rationalist Taboo - what question is this post even asking, when you remove explicit reference to the word "evidence"? You can't ask questions about a concept which you can't even identify.
I feel like I've seen an increasing amount of classical Traditional Rationality bullshit on this site lately, mostly in Discussion. That could just be me starting to notice it more, but I feel like I need to make a full post about it that I can link to whenever this stuff comes up. These are basic errors, explicitly warned against in the Sequences, and the whole point of Less Wrong is supposed to be somewhere where this sort of crap is avoided. Apologies for language.
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:
The second one says that e is evidence that h when the probability of h conditional on e is higher than some threshold 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 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:
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:
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?