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.
...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.
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”.
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.
I’m not sure if I understood what you said here. You agree with what I said in the first bullet or not?
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?
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.
“Defeat” would solely consist in the recognition of admitting to ~T instead of T. Not a matter of belief per se.
No, I don’t.
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.
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.