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Tyrrell_McAllister comments on Conceptual Analysis and Moral Theory - Less Wrong
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There is a reason why the Gettier rabbit-hole is so dangerous. You can always cook up an improbable counterexample to any definition.
For example, here is a counterexample to Nozick's definition as you present it. Suppose that I have irrationally decided to believe everything written in a certain book B and to believe nothing not written in B. Unfortunately for me, the book's author, a Mr. X, is a congenital liar. He invented almost every claim in the book out of whole cloth, with no regard for the truth of the matter. There was only one exception. There is one matter on which Mr. X is constitutionally compelled to write and to write truthfully: the color of his mother's socks on the day of his birth. At one point in B, Mr. X writes that his mother was wearing blue socks when she gave birth to him. This claim was scrupulously researched and is true. However, there is nothing in the text of B to indicate that Mr. X treated this claim any differently from all the invented claims in the book.
In this story, I am S, and P is "Mr. X's mother was wearing blue socks when she gave birth to him." Then:
P is true. (Mr. X's mother really was wearing blue socks.)
S believes that P. (Mr. X claimed P in B, and I believe everything in B.)
If it were the case that (not-P), S would not believe that P. (Mr. X only claimed P in B because that was what his scrupulous research revealed. Had P not been true, Mr. X's research would not have led him to believe it. And, since he is incapable of lying about this matter, he would not have put P in B. Therefore, since I don't believe anything not in B, I would not have come to believe P.)
If it were the case that P, S would believe that P. (Mr. X was constitutionally compelled to write truthfully about what the color of his mother's socks were when he was born. In all possible worlds in which his mother wore blue socks, Mr. X's scrupulous research would have discovered it, and Mr. X would have reported it in B, where I would have read it, and so believed it.)
And yet, the intuitions on which Gettier problems play would say that I don't know P. I just believe P because it was in a certain book, but I have no rational reason to trust anything in that book.
ETA: And here's a counterexample from the other direction — that is, an example of knowledge that fails to meet Nozick's criteria.
Suppose that you sit before an upside-down cup, under which there is a ping-pong ball that has been painted some color. Your job is to learn the color of the ping-pong ball.
You employ the following strategy: You flip a coin. If the coin comes up heads, you lift up the cup and look at the ping-pong ball, noting its color. If the coin comes up tails, you just give up and go with the ignorance prior.
Suppose that, when you flip the coin, it comes up heads. Accordingly, you look at the ping-pong ball and see that it is red. Intuitively, we would say that you know that the ping-pong ball is red.
Nonetheless, we fail to meet Nozick's criterion 4. Had the coin come up tails, you would not have lifted the cup, so you would not have come to believe that the ball is red, even if this were still true.
Yes. An excellent illustration of 'the Gettier rabbit-hole.'
Now, if only someone would give me a hand out of this rabbit-hole before I spend all morning in here ;).
Well, of course Bayesianism is your friend here. Probability theory elegantly supersedes the qualitative concepts of "knowledge", "belief" and "justification" and, together with an understanding of heuristics and biases, nicely dissolves Gettier problems, so that we can safely call "knowledge" any assignment of high probability to a proposition that turns out to be true.
For example, take the original Gettier scenario. Since Jones has 10 coins in his pocket, P(man with 10 coins gets job) is bounded from below by P(Jones gets job). Hence any information that raises P(Jones gets job) necessarily raises P(man with 10 coins gets job) to something even higher, regardless of whether (Jones gets job) turns out to be true.
The psychological difficulty here is the counterintuitiveness of the rule P(A or B) >= P(A), and is in a sense "dual" to the conjunction fallacy. Just as one has to remember to subtract probability as burdensome details are introduced, one also has to remember to add probability as the reference class is broadened. When Smith learns the information suggesting Jones is the favored candidate, it may not feel like he is learning information about the set of all people with 10 coins in their pocket, but he is.
In your example of the book by Mr. X, we can observe that, because Mr. X was constitutionally compelled to write truthfully about his mother's socks, your belief about that is legitimately entangled with reality, even if your other beliefs aren't.
I think that while what you define carves out a nice lump of thingspace, it fails to capture the intuitive meaning of the word probability. If I guess randomly that it will rain tomorrow and turn out to be right, then it doesn't fit intuition at all to say I knew that it would rain. This is why the traditional definition is "justified true belief" and that is what Gettier subverts.
You presumably already know all this. The point is that Tyrrell McAllister is trying (to avoid trying) to give a concise summary of the common usage of the word knowledge, rather than to give a definition that is actually useful for doing probability or solving problems.
I agree that, with regard to my own knowledge, I should just determine the probability that I assign to a proposition P. Once I conclude that P has a high probability of being true, why should I care whether, in addition, I "know" P in some sense?
Nonetheless, if I had to develop a coherent concept of "knowledge", I don't think that I'd go with "'knowledge' [is] any assignment of high probability to a proposition that turns out to be true." The crucial question is, who is assigning the probability? If it's my assignment, then, as I said, I agree that, for me, the question about knowledge dissolves. (More generally, the question dissolves if the assignment was made according to my prior and my cognitive strategies.)
But Getteir problems are usually about some third person's knowledge. When do you say that they know something? Suppose that, by your lights, they have a hopelessly screwed-up prior — say, an anti-Laplacian prior. So, they assign high probability to all sorts of stupid things for no good reason. Nonetheless, they have enough beliefs so that there are some things to which they assign high probability that turn out to be true. Would you really want to say that they "know" those things that just happen to be true?
That is essentially what was going on in my example with Mr. X's book. There, I'm the third person. I have the stupid prior that says that everything in B is true and everything not in B is false. Now, you know that Mr. X is constitutionally compelled to write truthfully about his mother's socks. So you know that reading B will legitimately entangle my beliefs with reality on that one solitary subject. But I don't know that fact about Mr. X. I just believe everything in B. You know that my cognitive strategy will give me reliable knowledge on this one subject. But, intuitively, my epistemic state seems so screw-up that you shouldn't say that I know anything, even though I got this one thing right.
ETA: Gah. This is what I meant by "down the rabbit-hole". These kinds of conversations are just too fun :). I look forward to your reply, but it will be at least a day before I reply in turn.
ETA: Okay, just one more thing. I just wanted to say that I agree with your approach to the original Gettier problem with the coins.
If you want to set your standard for knowledge this high, I would argue that you're claiming nothing counts as knowledge since no one has any way to tell how good their priors are independently of their priors.
I'm not sure what you mean by a "standard for knowledge". What standard for knowledge do you think that I have proposed?
You're talking about someone trying to determine whether their own beliefs count as knowledge. I already said that the question of "knowledge" dissolves in that case. All that they should care about are the probabilities that they assign to propositions. (I'm not sure whether you agree with me there or not.)
But you certainly can evaluate someone else's prior. I was trying to explain why "knowledge" becomes problematic in that situation. Do you disagree?
Here, let me introduce you to my friend Taboo...
;)