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.
I can't believe people apply Baye's theorem when confronted to counter-evidence. What evidence do we have to believe that Bayesian probability theories describe the way we reason inductively?
I present here two puzzles of rationality you LessWrongers may think is worth to deal with. Maybe the first one looks more amenable to a simple solution, while the second one has called attention of a number of contemporary epistemologists (Cargile, Feldman, Harman), and does not look that simple when it comes to a solution. So, let's go to the puzzles!
Puzzle 1
At t1 I justifiably believe theorem T is true, on the basis of a complex argument I just validly reasoned from the also justified premises P1, P2 and P3.
So, in t1 I reason from premises:
(R1) P1, P2 ,P3
To the known conclusion:
(T) T is true
At t2, Ms. Math, a well known authority on the subject matter of which my reasoning and my theorem are just a part, tells me I’m wrong. She tells me the theorem is just false, and convince me of that on the basis of a valid reasoning with at least one false premise, the falsity of that premise being unknown to us.
So, in t2 I reason from premises (Reliable Math and Testimony of Math):
(RM) Ms. Math is a reliable mathematician, and an authority on the subject matter surrounding (T),
(TM) Ms. Math tells me T is false, and show to me how is that so, on the basis of a valid reasoning from F, P1, P2 and P3,
(R2) F, P1, P2 and P3
To the justified conclusion:
(~T) T is not true
It could be said by some epistemologists that (~T) defeat my previous belief (T). Is it rational for me to do this way? Am I taking the correct direction of defeat? Wouldn’t it also be rational if (~T) were defeated by (T)? Why ~(T) defeats (T), and not vice-versa? It is just because ~(T)’s justification obtained in a later time?
Puzzle 2
At t1 I know theorem T is true, on the basis of a complex argument I just validly reasoned, with known premises P1, P2 and P3. So, in t1 I reason from known premises:
(R1) P1, P2 ,P3
To the known conclusion:
(T) T is true
Besides, I also reason from known premises:
(ME) If there is any evidence against something that is true, then it is misleading evidence (evidence for something that is false)
(T) T is true
To the conclusion (anti-misleading evidence):
(AME) If there is any evidence against (T), then it is misleading evidence
At t2 the same Ms. Math tells me the same thing. So in t2 I reason from premises (Reliable Math and Testimony of Math):
(RM) Ms. Math is a reliable mathematician, and an authority on the subject matter surrounding (T),
(TM) Ms. Math tells me T is false, and show to me how is that so, on the basis of a valid reasoning from F, P1, P2 and P3,
But then I reason from::
(F*) F, RM and TM are evidence against (T), and
(AME) If there is any evidence against (T), then it is misleading evidence
To the conclusion:
(MF) F, RM and TM is misleading evidence
And then I continue to know T and I lose no knowledge, because I know/justifiably believe that the counter-evidence I just met is misleading. Is it rational for me to act this way?
I know (T) and I know (AME) in t1 on the basis of valid reasoning. Then, I am exposed to misleading evidences (Reliable Math), (Testimony of Math) and (F). The evidentialist scheme (and maybe still other schemes) support the thesis that (RM), (TM) and (F) DEFEATS my justification for (T) instead. So that whatever I inferred from (T) is no longer known. However, given my previous knowledge of (T) and (AME), I could know that (MF): F is misleading evidence. It can still be said that (RM), (TM) and (F) DEFEAT my justification for (T), given that (MF) DEFEAT my justification for (RM), (TM) and (F)?