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 0 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.
The presumption of the claim "I know T is true" (and that evidence that it is false is false) is false precisely in the case that the reasoning used to show that T (in this case a theorem) is true is invalid. Were T not a theorem, then probabilistic reasoning would in fact apply, but it does not. (And since it doesn't, it is irrelevant to pursue that path. But, in short, the fact that it is a theorem should lead us to understand that the premisses' truth is not the issue at hand here, thus probabilistic reasoning need not apply, and so there is no issue of T's being probably true or false.) Furthermore, it is completely wide of the mark to suggest that one should apply this or that probability to the claims in question, precisely because the problem concerns deductive reasoning. All the non-deductive aspects of the puzzles are puzzling distractions at best. In essence, if a counterargument comes along demonstrating that T is false, then it necessarily would involve demonstrating that invalid reasoning was somewhere committed in someone's having arrived at the (fallacious) truth of T. (It is necessary that one be led to a true conclusion given true premisses.) Hence, one need not be concerned with the epistemic standing of the truth of T, since it would have clearly been demonstrated to be false. And to be committed to false statements as being not-false would be absurd, such that it would alone be proper to aver that one has been defeated in having previously been committed to the truth of T despite that that committment was fundamentally invalid. Valid reasoning is always valid, no matter what one may think of the reasoning; and one may invalidly believe in the validity of an invalid conclusion. Such is human fallibility.
So I'd say the problem is a wrong question.
No, I think it is a good question, and it is easy to be led astray by not recognizing where precisely the problem fits in logical space, if one isn't being careful. Amusingly (if not disturbingly), some of most up-voted posts are precisely those that get this wrong and thus fail to see the nature of the problem correctly. However, the way the problem is framed does lend itself to misinterpretation, because a demonstration of the falsity of T (namely, that it is invalid that T is true) should not be treated as a premiss in another apodosis; a valid demonstration of the falsity of T is itself a deductive conclusion, not a protasis proper. (In fact, the way it is framed, the claim ~T is equivalent to F, such that the claims [F, P1, P2, and P3] implies ~T is really a circular argument, but I was being charitable in my approach to the puzzles.) But oh well.
In essence, if a counterargument comes along demonstrating that T is false, then it necessarily would involve demonstrating that invalid reasoning was somewhere committed in someone's having arrived at the (fallacious) truth of T.
I think I see your point, but if you allow for the possibility that the original deductive reasoning is wrong, i.e. deny logical omniscience, don't you need some way to quantify that possibility, and in the end that would mean treating the deductive reasoning itself as bayesian evidence for the truth of T?
Unless you assume that...
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)?