Gemara

(Rav Yehudah and Rav Huna): If Reuven claims that Shimon owes to him 100 Zuz and Shimon says that he does not know, Shimon must pay. Reuven's Vadai (definite) claim is better than Shimon's doubtful claim.

Rav Nachman and R. Yochanan: Shimon is exempt. When in doubt, we leave money in its Chazakah (status quo; we do not force one to pay).

The Halachah follows Rav Nachman in monetary matters.

Bava Kama 118a (Mishnah): If Reuven tells Shimon 'I stole, received a deposit or borrowed from you, and I do not know whether I paid you', he is liable;

If Reuven is unsure whether or not he stole, received a deposit or borrowed, he is exempt.

Question (against Rav Huna - Seifa): If he is unsure whether or not he stole, received a deposit or borrowed, he is exempt.

Question: What is the case?

Answer #1: Shimon does not claim from him.

Rejection: If so, we must say so also in the Reisha. Why is Reuven liable?!

Answer #2: Rather, he claims from him, and the Seifa says that he is exempt.

Answer (and defense of Answer #1): Really, Shimon does not claim from him. The Reisha discusses when Reuven wants to be Yotzei Yedei Shomayim (fulfill his obligation to Heaven).

(R. Chiya bar Aba citing R. Yochanan): If Levi tells Yehudah 'You owe to me 100', and Yehudah says that he does not know, if he wants to be Yotzei Yedei Shomayim he is liable.

Rishonim

Rif (Kesuvos 4b): We must say that Rav Nachman holds like R. Gamliel to avoid a contradiction. The Halachah follows Rav Nachman in monetary matters, and the Halachah follows R. Gamliel in our Mishnah!

Rif (Bava Kama 44b): The Mishnah (in Bava Kama) is only when Shimon claims from Reuven. Since Reuven knows that he stole or borrowed and is unsure whether or not he paid, he is obligated, for a Safek does not prevail against a Vadai. If Reuven said this by himself and Shimon does not claim, Reuven pays only if he wants to be Yotzei Yedei Shomayim. If Shimon claims and Reuven is unsure whether or not he borrowed, he is exempt, like our Mishnah. Rav Huna establishes our Mishnah to be when Shimon does not claim. The Halachah does not follow him. Rather, it is like Rav Nachman. Even if Shimon claims, Reuven is exempt. He must swear that he does not know, and if he wants to be Yotzei Yedei Shomayim he must pay.

Nimukei Yosef (DH Garsinan): When Reuven is unsure whether or not he ever owed he must swear. This is no better than if he said that he Vadai does not know, in which case he must swear Heses. This is like Rashi and the Rosh, who say that we follow the version that obligates Shevu'as Heses even without Drara (something substantiating a claim) of money. Rav Hai Gaon and R. Tam impose Heses only when there is Drara.

Ran (Kesuvos 4b DH Gemara): If Reuven need not swear (when he says that he is unsure whether or not he ever owed), there will never be Shevu'as Heses!

Note: It seems that the Ran holds like Rashi and the Rosh, and teaches that if we exempt one who claims to be unsure, everyone will claim this.

Gra (CM 75:25): We enacted Heses because even though a borrower normally does not deny a debt, normally a person does not make an empty claim. All the more so it applies here, for we assume that Shimon would not make an empty claim, and Reuven does not (Vadai) deny it!

Rambam (Hilchos To'en 1:8): If Shimon claims that Reuven definitely owes to him 100 Zuz and Reuven says that he does not know, Reuven takes a (mid'Rabanan) oath called Heses that he does not know, and he is exempt, for he did not definitely obligate himself. The same applies to all similar cases.

Hagahos Maimoniyos (90): R. Shimshon, Sefer ha'Mitzvos, Avi ha'Ezri and other Ge'onim obligate Shevu'as Heses.

Rambam (9): If Shimon claims that Reuven definitely owes to him 100 Zuz and Reuven says that he owed but does not know whether he paid, Reuven must pay. Shimon does not swear at all, even Heses, for he claims Vadai and Reuven admits, just Reuven is unsure whether or not he paid. If Shimon does not claim, but Reuven himself admitted that he borrowed or stole from Shimon or took a deposit from Shimon's father (who later died), and Reuven is unsure whether or not he paid, he is exempt. If he wants to be Yotzei Yedei Shomayim he must pay.

Question (Magid Mishneh 9): Why didn't the Rambam say in Halachah 8 that Reuven must pay to be Yotzei Yedei Shomayim?

Answer (Shiltei ha'Giborim Shevu'os Reish 20a): First the Rambam clarified when Beis Din forces one to pay. Afterwards he said that one must pay to be Yotzei Yedei Shomayim. It is clear that this refers to every case when he is exempt b'Yedei Adam.

Mishneh l'Melech: When Poskim argue about whether or not something is considered payment, the defendant can say 'I hold like the opinion that it is payment.' Lechem Rav (176) disagrees, for he knows that he was liable but is unsure whether or not he paid. This is wrong.

Rosh (Bava Kama 10:32): If Shimon claims that Reuven owes to him 100 Zuz and Reuven says that he does not know, Reuven is exempt, for the Halachah follows Rav Nachman. He swears that he does not know. If Reuven is unsure whether or not he paid, Rav Nachman agrees that he must pay. We learn from the Gemara. It challenged Rav Huna from the Seifa, assuming that this is when Shimon claims, like the case in which Rav Huna and Rav Nachman argue. (In the Reisha, Reuven admits that he owed and is unsure whether or not he paid, and he is liable. The Gemara did not challenge anyone from the Reisha, for all agree to it.)

Lechem Mishneh (Hilchos To'en 9): The Gemara said that R. Chiya bar Aba's law (if Levi claims from Yehudah and Yehudah is unsure, he must pay to be Yotzei Yedei Shomayim) is similar to how Rav Huna establishes our Mishnah. I.e. in both cases there is a Vadai claim and one is unsure whether or not he (still) owes. Therefore, in both cases he must pay to be Yotzei Yedei Shomayim.

Rosh (ibid.): In summary, when Shimon claims and Reuven admits that he owed and is unsure whether or not he paid, all agree that he is liable. If Reuven is unsure whether or not he paid, he is exempt, but he must swear. To be Yotzei Yedei Shomayim he must pay;

If Shimon does not claim and Reuven admits that he owed and is unsure whether or not he paid, it seems that even Rav Nachman agrees that he must pay to be Yotzei Yedei Shomayim. If Reuven is unsure whether or not he ever owed, he is exempt even b'Yedei Shomayim.

Poskim

Shulchan Aruch (CM 75:9): If Shimon claims 'I lent to you or deposited with you 100 Zuz' and Reuven is unsure whether he took a loan or deposit, Reuven swears Heses and he is exempt. If he wants to be Yotzei Yedei Shomayim he must pay. Even if Reuven initially denied it, and when asked to swear Heses said that he is unsure, he swears that he does not know.

Shach (25): We do not distinguish whether Shimon claims that Reuven borrowed, took a deposit or stole.

Shach (26): The Rivash (392) cites the Ramban (Bava Metzi'a 5b DH mideka'Amrinan) to say that if witnesses saw Reuven take what he claims from Shimon, he must return it. Seizure does not help when the defendant only 'owes' to be Yotzei Yedei Shomayim.

Beis Yosef (DH Kosav): Sefer ha'Terumos (38:3:5) and Rabbeinu Yehudah say that if one witness says that Reuven borrowed 100 and Reuven is unsure whether or not he borrowed, since he must swear (to contradict the witness) but he cannot, he must pay. Others exempt, for he is not considered to be one who must swear, since he is unsure. This is wrong.

Shulchan Aruch (ibid.): If Shimon claims that he lent or deposited 100 Zuz to or with Reuven, and Reuven admits but is unsure whether he returned it, he must pay.

Shach (27): This is even if Reuven has a Safek whether he 'paid' before he ever owed. E.g. Shimon owed him for negligence with a deposit and says that it was worth only 50, and Reuven says that perhaps it was worth 100, and afterwards Reuven borrowed 100 from Shimon.

Shulchan Aruch (ibid.): Shimon does not swear at all, even Heses. Reuven may impose a Cherem on anyone who fraudulently takes his money.

Shach (28): One does not swear Heses due to a Safek claim.

Rema: If Reuven later says that he remembered that he paid, he is believed through an oath.

Yam Shel Shlomo (Bava Kama 10:55, brought in Shach 29): This is only according to Sha'arei Rav Alfas, who requires Shimon to swear (when Reuven is unsure whether or not her paid), for it was not clear that (Shimon will swear and) Reuven needs to pay. Therefore, we do not say that Reuven changed his claim. We hold that Shimon need not swear, so Reuven cannot later say that he paid to exempt himself.

Rebuttal (Gra 29): Even though he changes his claim from liable to exempt, this is not a retraction, for he does not contradict himself.