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.
at least one of your premises is logically inconsistent with he others—that they cannot all be true.
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.
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.