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.