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Michael_Sullivan comments on two puzzles on rationality of defeat - Less Wrong Discussion
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Confidence that the same premises can imply both ~T and T is confidence that at least one of your premises is logically inconsistent with he others -- that they cannot all be true. It's not just a question of whether they model something correctly -- there is nothing they could model completely correctly.
In puzzle one, I would simply conclude that either one of the proofs is incorrect, or one of the premises must be false. Which option I consider most likely will depend on my confidence in my own ability, Ms. Math's abilities, whether she has confirmed the logic of my proof or been able to show me a misstep, my confidence in Ms. Math's beliefs about the premises, and my priors for each premise.
Suppose I have three axioms: A, B, and C.
A: x=5
B: x+y=4
C: 2x+y=6
Which axiom is logically inconsistent with the others? (A, B), (B, C), and (A, C) are all consistent systems, so I can't declare any of the axioms to be false, just that for any particular model of anything remotely interesting, at least one of them must not apply.