Rambam - 1 Chapter a Day

As explained in Chapter 18, Halachah 2, a contradiction between two groups of witnesses refers to a situation where one group claims that an event took place, e.g., a murder was committed or a loan was given, and another group claims that the event never took place.

Since the testimony they originally gave contradicted each other, one of them was certainly not telling the truth. That person is not acceptable as a witness. Although we do not know which one - and hence cannot label either of them as a lying witness - we know that one of them is unacceptable. And thus when they come together, we are certain that at least one of the witnesses is unacceptable.
The Sefer Meirat Einayim 31:2 explains that the defendant is not even required to take an oath, as would be required when one witness testifies in support of the plaintiff. The rationale is that when one of a group of witnesses is disqualified, the entire testimony is of no consequence (Chapter 5, Halachah 3).

Since we do not know which of the groups is lying, we do not disqualify either of them. Instead, we follow the basic assumption that every Jew is acceptable as a witness (Sh’vuot 47b).

100 zuz.

I.e., the lesser sum of the two amounts.

For he is trying to expropriate funds from the person in possession (see Hilchot Malveh ViLoveh 27:15-16). Were the promissory notes to be signed by acceptable witnesses, he would be in a position of strength. In this instance, however, since one of the groups of witnesses certainly lied and is hence disqualified, the plaintiff is put at a disadvantage.

After the defendant makes the smaller payment and takes the oath, the larger promissory note is destroyed by the court, so that it could not be used again [Shulchan Aruch (Choshen Mishpat 31:2)].
The Shulchan Aruch continues, stating that the Rambam’s ruling applies only in a situation where the plaintiff demands payment for both promissory notes at the same time. If, however, he demands payment for each one individually, his claim is vindicated and the defendant is required to pay both the debts. See Halachah 4.
The Ramah adds that the defendant may require the plaintiff to take an oath that he does not possess two promissory notes. For this reason, he suggests that the plaintiff should burn the smaller promissory note so that he will at least receive payment for the larger one.

I.e., holding a Torah scroll or tefillin, as stated in Hilchot Sh’vuot 11:8.

I.e., this is a Scriptural oath, and not merely a sh’vuat hesset of Rabbinic origin.

For at least one of the groups of witnesses is acceptable.

As explained in Hilchot To’en V’Nit’an 4:10, when a plaintiff demands 100, the defendant denies the obligation entirely, and witnesses testify that he owes 50, the defendant is required to take a Scriptural oath concerning the remaining 50. The rationale is that just as he would be required to take such an oath had he admitted owing the 50 himself, he is required to take the oath when his obligation is established by the testimony of witnesses.
The Ra’avad and Rabbenu Nissim object to this ruling, explaining - as the Rambam states in Hilchot Toen ViNitan 4:4 - that when a person acknowledges a debt recorded in a promissory note, the laws concerning one who admits a portion of a debt do not apply to him. For even were he to deny the debt in this instance, his denial would not be accepted. Similarly, in the case at hand, one could claim that since there is a promissory note involved, there should be no obligation for an oath. Also, a promissory note establishes a lien against landed property and in such instances, an oath is not required.
A distinction can, however, be made. For in this instance, neither of the promissory notes have the power to enable the plaintiff to expropriate property on its basis. Hence the claim resembles one supported by a verbal commitment alone (see Kessef Mishneh, Radbaz). The Siftei Cohen 31:2, however, does not accept these resolutions and states that the Rambam’s ruling applies only when the defendant does not own land.

Since we do not know which of the claims is valid and which is using lying witnesses, we substantiate them both.

See Hilchot Malveh v’Loveh 16:5. As the Rambam states there, this oath was instituted to appease the defendant. Since he is obligated to pay two claims and one of them is not legitimate, he is at least given the satisfaction of requiring both of the plaintiffs to take oaths.

Because one - and we do not know which one - is based on the testimony of lying witnesses.

I.e., of no value.

A less severe oath required by our Sages when a defendant denies entirely a claim lodged against him.

For each of the defendants can say: “It is the pair of witnesses on the other document that are acceptable. Those who signed my document are liars.” Since we don’t know which of the defendants’ claims to accept - and one is certainly true - we decide in favor of both defendants. For whenever a person is trying to expropriate money, the burden of proof is upon him.

Rabbenu Asher and the Tur rule that the plaintiff must conduct the suits in two different courts. The Shulchan Aruch (Choshen Mishpat 31:3) quotes the Rambam’s ruling without making this clarification.

For as stated in Halachah 1, when each one of the groups comes individually, their testimony is accepted.

I.e., we look at the witnesses as we look at any group of witnesses, assuming that they are telling the truth.

For we assume that the plaintiff indeed had the legal document forged.

As the Rambam states in Chapter 8, Halachot 1 and 4, when witnesses testify with regard to their signature on a legal document, they must testify that the transaction in fact took place. Hence, we rely on their word even if the plaintiff is known to have attempted to perpetrate forgery.
The Radbaz states that even if the witnesses do not remember signing the legal document, since they remember the loan and recognize their signatures, the document is valid. We do not suspect it is a forgery.