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Michael_Sullivan comments on two puzzles on rationality of defeat - Less Wrong
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Comments (55)
Axioms are not true or false. They either model what we intended them to model, or they don't. In puzzle 1, assuming you have carefully checked both proofs, confidence that (F, P1, P2, P3) implies T and (F, P1, P2, P3) implies ~T are both justified, rendering (F, P1, P2, P3) an uninteresting model that probably does not reflect the system that you were trying to model with those axioms. If you are trying to figure out whether or not T is true within the system you were trying to model, then of course you cannot be confident one way or the other, since you aren't even confident of how to properly model the system. The fact that your proof of T relied on fewer axioms would seem to be some evidence that T is true, but is not particularly strong.
puzzle 2: (ME) points both ways. While it certainly seems to be strong evidence against the reliability of (RM), since she just reasoned from clearly inconsistent axioms, it can't prove that F is the axiom you should throw away. Consider the possibility that you could construct a proof of ~T given only F, P1, and P2. Now, (ME) could not possibly say anything different about F and P3.
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.