C1 is a presumption, namely, a belief in the truth of T, which is apparently a theorem of P1, P2, and P3. As a belief, it's validity is not what is at issue here, because we are concerned with the truth of T.

F comes in, but is improperly treated as a premiss to conclude ~T, when it is equivalent to ~T. Again, we should not be concerned with belief, because we are dealing with statements that are either true or false. Either but not both (T or ~T) can be true (which is the definition of a logical tautology).

Hence C2 is another presumption with which we should not concern ourselves. Belief has no influence on the outcome of T or ~T.

For the first bullet: no, it is not possible, in any case, to conclude C2, for not to agree that one made a mistake (i.e., reasoned invalidly to T) is to deny the truth of ~T which was shown by Ms. Math to be true (a valid deduction).

Second bullet: in the case of a theorem, to show the falsity of a conclusion (of a theorem) is to show that it is invalid. To say there is a mistake is a straightforward corollary of the nature of deductive inference that an invalid motion was committed.

Third bullet: I assume that the problem is stated in general terms, for had Ms. Math shown that T is false in explicit terms (contained in F), then the proper form of ~T would be: F -> ~T. Note that it is wrong to frame it the following way: F, P1, P2, and P3 -> ~T. It is wrong because F states ~T. There is no "decision" to be made here! Bayesian reasoning in this instance (if not many others) is a misapplication and obfuscation of the original problem from a poor grasp of the nature of deduction.

(N.B.: However, if the nature of the problem were to consist in merely being told by some authority a contradiction to what one supposes to be true, then there is no logically necessity for us to suddenly switch camps and begin to believe in the contradiction over one's prior conviction. Appeal to Authority is a logical fallacy, and if one supposes Bayesian reasoning is a help there, then there is much for that person to learn of the nature of deduction proper.)

Let me give you an example of what I really mean:

Note statements P, Q, and Z:

(P) Something equals something and something else equals that same something such that both equal each other. (Q) This something equals that. This other something also equals that. (Z) The aforementioned somethings equal each other.

It is clear that Z follows from P and Q, no? In effect, you're forced to accept it, correct? Is there any "belief" involved in this setting? Decidedly not. However, let's suppose we meet up with someone who disagrees and states: "I accept the truths of P and Q but not Z."

Then we'll add the following to help this poor fellow:

(R) If P and Q are true, then Z must be true.

They may respond: "I accept P, Q, and R as true, but not Z."

And so on ad infinitum. What went wrong here? They failed to reason deductively. We might very well be in the same situation with T, where

(P and Q) are equivalent to (P1, P2, and P3) (namely, all of these premisses are true), such that whatever Z is, it must be equivalent to the theorem (which would in this case be ~T, if Ms. Math is doing her job and not merely deigning to inform the peons at the foot of her ivory tower).

P1, P2, and P3 are axiomatic statements. And their particular relationship indicates (the theorem) S, at least to the one who drew the conclusion. If a Ms. Math comes to show the invalidity of T (by F), such that ~T is valid (such that S = ~T), then that immediately shows that the claim of T (~S) was false. There is no need for belief here; ~T (or S) is true, and our fellow can continue in the vain belief that he wasn't defeated, but that would be absolutely illogical; therefore, our fellow must accept the truth of ~T and admit defeat, or else he'll have departed from the sphere of logic completely. Note that if Ms. Math merely says "T is false" (F) such that F is really ~T, then the form [F, P1, P2, and P3] implies ~T is really a circular argument, for the conclusion is already assumed within the premisses. But, as I said, I was being charitable with the puzzles and not assuming that that was being communicated.