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On Bava Metzia 3a, it talks about a concept that if a plaintiff claims Bob owes him 100 and Bob says no I don’t, nothing is done, but if Bob admits and says “I only only you 50, not 100,” (or if witnesses say he owes at least 50) then Bob has to swear that he doesn’t owe the other 50.

In mishnayot Ketubot, there’s this concept of הפה שאסר הוא הפה שהתיר. For example, if a woman says “I was taken captive but am pure,” we believe her because she gave herself the thought of being impure by telling us she was a captive, so therefore the latter half of her statement is believable (otherwise if she was impure, she wouldn’t have mentioned being captive in the first place to take away suspicion). But if witnesses say “she was taken captive” and she says “no I’m pure,” we now don’t believe her because she didn’t אסר herself.

Anyway, while these two things don’t exactly line up, my question is why in the case of when Bob admits he owes 50 do we not believe him about the other 50 and we make him swear? Him admitting to owing 50 seems to be a case of הפה שאסר and you would think that we believe he doesn’t actually owe the other 50, because otherwise he would’ve just said he owes nothing and then he’d be fine and have no suspicion against him. So why doesn’t the concept of הפה שאסר apply to the Bava Metzia case?

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    That’s essentially Rabbah’s question at the bottom of the amud - מפני מה אמרה תורה דמודה מקצת הטענה ישבע
    – Joel K
    Commented Nov 19, 2023 at 18:00

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While it's a little more complicated than this (and is, if I remember correctly, a machlokes Rishonim in how to understand both of these rules), this is more or less the difference.

"הפה שאסר הוא הפה שהתיר" is said in a case where we would have no reason to be suspicious of the defendant. This is caused through a מיגו, meaning we say "if the defendant had wanted to lie, he could have done/ said xyz, which would be more advantageous for him" (more or less. Again, argument in the rishonim).

For example, if a woman says "I was taken captive but I am pure" we believe her; because she is the one that caused us to be suspicious in the first place, she can alleive that suspicion with her own words (in other words, if she were going to lie in order to get out of the situation, she wouldn't have said anything at all. This is called a מיגו דאי בעיה שתיק, meaning "from the fact that he could have been quiet had he wanted to").

On the other hand, we do have reason to be suspicious of the defendant (Bob, as per OP's example) in the case in Bava Metzia. Firstly, because he is being sued and not himself raising the suspicion (though tosafos on Kesubos 15b says this doesn't affect the ruling, either way הפה שאסר should be applicable, Rashi there disagrees). And secondly, more importantly, in Bava Metzia 3a - 3b the defendant has no מיגו. We are suspicious of him, because this is exactly what he would say if he were lying:

דאמר רבה: מפני מה אמרה תורה מודה מקצת הטענה ישבע חזקה אין אדם מעיז פניו בפני בעל חובו והאי בכוליה בעי דנכפריה והא דלא כפריה משום שאין אדם מעיז פניו בפני בעל חובו

As Rabbah said: why did the Torah say that one who admits to owing part of the claim must swear? We have a Chazaka that a man doesn't brazenly lie in front of his lender. And really he would like to deny owing anything, and the fact that he didn't is because a man doesn't brazenly lie in front of his lender

(The translation is mine, sorry for any bad grammar etc.)

In essence, he is too scared or embarrassed to completely deny his debt, so he admits to owing half (the gemara discusses in the next line or so why he isn't too embarrassed to only admit to half). Which means, that since there is no more advantageous lie he could feasibly claim, we entertain the possibility that he is lying, and we don't say "הפה שאסר הוא הפה שהתיר"

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