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SUMMARY
A person who is Mafkir his property may only back out if he was Mafkir it in front of two people even according to Rebbi Yosi. (1)
Rebbi Yochanan says that if someone is Mafkir in front of three people it is Hefker, but not if he was Mafkir in front of two people.
Rebbi Yehoshua Ben Levi says that Min ha'Torah if someone is Mafkir something in front of one person it is Hefker. (2)
If two partners make a Neder not to benefit from each other they may not walk into their shared courtyard. (3)
Rebbi Eliezer Ben Yakov holds that they are permitted to walk into their shared courtyard.
A BIT MORE
1. When someone is Mafkir in front of two people the Hefker does not have a Kol and therefore he must have been Mafkir it with one of those two people in mind and consequently it is comparable to a gift. However, if he was Mafkir it in front of three people it does have a Kol and therefore he was Mafkir it with the entire world in mind and thus it is not similar to a gift and he may not back out.
2. However the Rabanan said that you must be Mafkir it in front of three people so that if one of those three is Zocheh in it there will be two witnesses to testify that he was Zocheh.
3. This Tana holds that Vitur (insignificant benefit) is forbidden for someone who vowed not to benefit from his friend and therefore it is forbidden to even walk into the courtyard.
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BRIEF INSIGHT
BITUL CHAMETZ
Rebbi Yehoshua Ben Levi says that Min ha'Torah if someone is Mafkir something in front of one person it is Hefker. The Rosh says that one person is Lav Davka and not only if a person is Mafkir it in front of one person is it Hefker but even if he is Mafkir it in seclusion without anyone present it is a valid Hefker Min ha'Torah. However the Rambam disagrees and he holds that there must be at least one person present when one is Mafkir it. The Shalmei Nedarim asks according to the Rambam who holds that there must be one person present why is a person allowed to do Bitul Chametz on Erev Pesach without anyone present. Bitul Chametz works in the same way as Hefker according to many Mefarshim. The Shalmei Nedarim answers that the Ran says in Pesachim that the Hefker of Chametz does not require all of the Dinim of Hefker because once it becomes Asur b'Hana'ah on Erev Pesach it is no longer in his possession, however the Torah still requires him to do Bitul because the Torah places it in his possession with respect to Bal Yera'eh.
QUICK HALACHAH
HEFKER
If someone is Mafkir his land the first person to make a Kinyan on it is Zocheh even if he was Mafkir it in front of only one person. When a person is Zocheh in it from Hefker it is Patur from Ma'aser. However mid'Rabanan it is not Hefker unless he is Mafkir it in front of three people so that if one if them is Zocheh in it the other two people are witnesses. According to some opinions even if there was no one present when he was Mafkir it, it is Hefker. (Shulchan Aruch CM 273:7)
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