Gust comments on two puzzles on rationality of defeat - Less Wrong

4 Post author: fsopho 12 December 2011 02:17PM

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Comment author: Gust 13 December 2011 11:18:11PM 0 points [-]

I guess it wasn't clear, C1 and C2 reffered to the reasonings as well as the conclusions they reached. You say belief is of no importance here, but I don't see how you can talk about "defeat" if you're not talking about justified believing.

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).

I'm not sure if I understood what you said here. You agree with what I said in the first bullet or not?

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.

Are you sure that's correct? If there's a contradiction within the set of axioms, you could find T and ~T following valid deductions, couldn't you? Proving ~T and proving that the reasoning leading to T was invalid are only equivalent if you assume the axioms are not contradictory. Am I wrong?

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.

The problem I see here is: it seems like you are assuming that the proof of ~T shows clearly the problem (i.e. the invalid reasoning step) with the proof of T I previously reasoned. If it doesn't, all the information I have is that both T and ~T are derived apparently validly from the axioms F, P1, P2, and P3. I don't see why logic would force me to accept ~T instead of believing there's a mistake I can't see in the proof Ms. Math showed me, or, more plausibly, to conclude that the axioms are contradictory.

Comment author: argumzio 14 December 2011 03:41:39AM 2 points [-]

...I don't see how you can talk about "defeat" if you're not talking about justified believing

"Defeat" would solely consist in the recognition of admitting to ~T instead of T. Not a matter of belief per se.

You agree with what I said in the first bullet or not?

No, I don't.

The problem I see here is: it seems like you are assuming that the proof of ~T shows clearly the problem (i.e. the invalid reasoning step) with the proof of T I previously reasoned. If it doesn't, all the information I have is that both T and ~T are derived apparently validly from the axioms F, P1, P2, and P3.

T cannot be derived from [P1, P2, and P3], but ~T can on account of F serving as a corrective that invalidates T. The only assumptions I've made are 1) Ms. Math is not an ivory tower authoritarian and 2) that she wouldn't be so illogical as to assert a circular argument where F would merely be a premiss, instead of being equivalent to the proper (valid) conclusion ~T.

Anyway, I suppose there's no more to be said about this, but you can ask for further clarification if you want.

Comment author: Gust 14 December 2011 04:35:54AM 1 point [-]

2) that she wouldn't be so illogical as to assert a circular argument where F would merely be a premiss, instead of being equivalent to the proper (valid) conclusion ~T.

Oh, now I see what you mean. I interpreted F as a new promiss, a new axiom, not a whole argument about the (mistaken) reasoning that proved T. For example, (wikipedia tells me that) the axiom of determinacy is inconsistent with the axiom of choice. If I had proved T in ZFC, and Ms. Math asserted the Axiom of Determinacy and proved ~T in ZFC+AD, and I didn't know beforehand that AD is inconsistent with AC, I would still need to find out what was the problem.

I still think this is more consistent with the text of the original post, but now I understand what you meant by " I was being charitable with the puzzles".

Thank you for you attention.

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