Gemara

(Mishnah): An oath about something impossible, e.g. 'if I did not see a camel fly, is a Shevu'as Shav (vain oath).

Question (Ravina): Perhaps he saw a huge bird, nicknamed it 'camel' and swore truthfully based on his Da'as (intent of his words)!

Answer: We judge oaths (objectively) based on the words he said, and not based on his personal intent.

Question #1 (Beraisa): When Beis Din administers an oath, we tell the one who must swear 'you do not swear based on your own Da'as, rather based on the Da'as (understanding) of Hash-m and Beis Din.'

Suggestion: If Beis Din did not say this, he could swear based on his Da'as, e.g. he could nickname some wood chips 'money', and swear 'I gave the money to Ploni'!

Answer: No. (Really, we judge oaths objectively based on his words, and not based on his Da'as. We are concerned for cases like the following._

Nedarim 25a: Reuven claimed that he paid his debt to Shimon. Rava told him to swear. Reuven was using a hollow cane with coins idden inside. Reuven asked Shimon to take the reed, held a Sefer Torah and swore that he paid Shimon. Upon hearing the oath, Shimon broke the reed in anger. The coins fell out, and it was seen that the oath was true.

Question #2 (Beraisa): Moshe forced Bnei Yisrael to swear. He said 'you do not swear based on your Da'as, rather, on Hash-m's Da'as and mine.'

Had he made them swear to observe all the Mitzvos, this could be construed to mean Tzitzis, which is equivalent to the entire Torah.

Counter-question: Why wasn't it enough to make them swear on Moshe's Da'as? Why did he require also on Hash-m's Da'as'?

Answer to both questions: This is in order that the oath (will be based on the Da'as of many (others), so it) cannot be annulled.

Gitin 35a: A widow came in front of Rav Huna to collect her Kesuvah. Rav Huna said 'I hold like Rav, who does not collect a Kesuvah for a widow.'

The woman: You suspect lest I already received part of my Kesuvah. I swear by Hash-m's name that I did not benefit from my Kesuvah!

(Rav Huna): Rav admits that if she swears on her own, she receives her Kesuvah.

Nedarim 28a (Mishnah): One may falsely vow to extortionists that the property belongs to the king.

(Rav Amram): He says '(all) Peros are forbidden to me today if this is not the king's'. He says 'today' only in his heart, not with his lips.

Even though Devarim sheb'Lev (unspoken conditions) are ignored, in the case of Ones they are valid.

Bava Metzia 2a (Mishnah): If two are holding a Talis, and both say 'it is mine', each swears that he does not own less than half, and they divide it.

If Reuven says 'it is all mine' and Shimon says 'it is half mine', Reuven swears that he does not own less than three quarters, and Shimon swears that he does not own less than one quarter. Each gets like he swore.

Rishonim

Rif and Rosh (3:21): Our Sugya shows that we do not follow his Da'as. Since no one else calls a bird 'camel', his opinion is Batel to everyone else's, and his oath was vain.

Rambam (Hilchos Shevuos 2:15): One who swears 'I will not eat today. I swear on your Da'as' cannot say 'this is what I meant', for he did not swear on his own Da'as, rather, on that of others. Since his mouth and the hearts of the others were the same, their hearts (intent) is in place of his heart. The same applies to other kinds of Shevuos.

Kesef Mishneh: This is reasonable. He learns from 29a, 'we do not make you swear based on your own Da'as.'

Rambam (16): When Beis Din administers an oath, we say 'we do not make you swear based on your own Da'as, rather, based on our Da'as.'

Rosh (Bava Metzia 1:1): Letter of the law, Reuven should receive, without swearing, half the Talis that Shimon does not claim. We force him to swear that he owns no less than three quarters, lest he scheme. Even though he swears Al Da'as Beis Din, we enact what is possible to avoid scheming. This shows that if a defendant (Ploni) is scheming, the judge can be stringent and specify every possible intent that Ploni could have in the oath. Ploni cannot say 'there is no need for this. I swear according to your Da'as, without scheming.' One fears more when they mention how he transgresses the oath, than when he swears Stam, even though according to the judge's Da'as he transgresses.

Maharit (Sof 70): A case occurred in which the parties contradicted each other, and Moshe (the Muchzak) could not swear. He cannot demand 'let Levi swear that I have no claim on any of it.' Even if Moshe, based on his initial claim, is sure that 'Levi swears falsely, and my claim is correct', we do not force Levi to specify his oath. If we would, we show that we believe Moshe, for he is Muchzak! Rashi explains that Abaye says that when both hold the Talis and claim all of it, they swear lest one of them is owed an old debt, and he knows that the other forgot and will deny it, therefore, he grabbed onto his Talis and is ready to swear that half is his, for he has a lien on it. If so, all the more so we should be concerned (lest Levi swear that it is all his, when it was his and Moshe seeks to take it for a debt). Even so, I say that the judge should not specify, lest a litigant learn from this how to claim falsely. We let Levi swear Stam. Likewise, we do not suggest Gilgul Shevu'ah. The Gemara said that we make one swear 'I own in it, and I do not own less than half', lest one intend 'I swear Emes, that I do not own any of it.' Tosfos says that even though one swears Al Da'as Beis Din, it is good to specify.

Poskim

Shulchan Aruch (CM 87:20): We tell him 'we do not make you swear based on your own Da'as, rather, based on our Da'as and Da'as Beis Din.'

Beis Yosef (DH v'Im): The Beraisa mentions 'Al Da'as Hashem', but the Rambam and Tur mention only 'our Da'as.'

Rema: If there is an angle of trickery, the judge should specify every trickery that the person could be thinking.

SMA (62): In this case, it is not enough that one swears Al Da'as Beis Din.

Beis Yosef (DH Motzasi, citing Teshuvas Maharam 337): If Ploni swore on his own to Shimon before he was obligated, if later, Beis Din obligated him to swear, he was not Yotzei with his first oath. One must swear after Beis Din rules. Perhaps he stole after he swore, or he called wood chips 'money', or it is a case like the stick in Rava's Beis Din. Rav admits that if a widow swore on her own, she collects her Kesuvah (Gitin 35a). Really, she swore Al Da'as Beis Din; the Gemara did not elaborate for something that is not needed. R. Yerucham says in the name of R. Tam that if one swore on his own, he swears again Al Da'as Beis Din.

Shulchan Aruch (YD 210:1): If one vowed Al Da'as of others, it does not depend on what he thought. The vow takes effect based on their Da'as.

Gra (3): He comes to answer the Meforshim's question from Nedarim 28a, that Devarim sheb'Lev Einam Devarim. In Shevuos, we say that one cannot give a Perush far from the simple meaning, and if he explicitly said 'Al Da'atan (of Beis Din)', he cannot give any Perush! The Rambam discusses such a case. Normally, one can say that there was a Tanai, like R. Akiva did. (Maseches Kalah records that he swore to bring a woman to the world to come, so she would help him to clarify a Halachah, and he annulled his oath while saying it, in his heart.) All the more so one who vows on his own Da'as may say that there was a Tanai. A Tosefta (Taharos 6:9) explicitly says so.

Beis Lechem Yehudah (3): This is not only for one who vows Al Da'as of others. Also if Ploni swears at the request of David and for David's benefit, even if he did not swear Al Da'as David, it does not depend on Ploni's heart (Maharashdam YD 112). If one said 'I swear Al Da'as Rabim', and later said that he did not mean this and merely said it due to habit, he is not believed (Tumas Yesharim 141). If Ploni swore to David 'whatever I collect from this debt, I will give to you half or a third', and he did not collect it, rather, he sold it for less than its value, he must pay David according to the value (Ra'anach).