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In the Torah, sometimes it says, "two or three witnesses" are needed; why not simply say "two witnesses" are needed, and leave it at that?

Is it because the Torah is allowing judges, based on their discretion, to set the needed witnesses to the higher number of three, for some cases, whereas other judges, based on their discretion, might just ask for two?

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It is to teach some rules that refer to the use of witnesses when there are more than two witnesses present. For example, Babylonian Talmud: Tractate Sanhedrin Folio 9a

And just as in the case of two witnesses, if one is found to be a near kinsman or otherwise disqualified4 person, the whole testimony is rendered void, so in the case of three witnesses, the disqualification of one invalidates the whole evidence. And whence do we infer that this law would apply even if the number of witnesses reached a hundred? — We infer it from the repetition of the word witnesses.

The talmud explains that the group of witnesses is treated as a single unit and once they appear before the court, a disqualified witness will invalidate the entire testimony.

Art Scroll Sanhedrin 9a note 8 says

If three witnesses are discredited through hazamah (see 8b note 2) they are all punished equally, even though the court could have ruled on the strength of the testimony of the first two witnesses alone

Note 13

The verse states By the word of two witnesses or three witnesses, where it would have sufficed to say , two or three witnesses. The repetition of the word witnesses teaches that all groups of witnesses, whatever their size, are treated equally.

An example of such a case is found in Business halacha Weekly

“Asher is your cousin?” asked Yankel.

“Yes, our fathers are brothers,” said Yitzi.

“If so, he is disqualified from testifying,” said Yankel. “A first cousin is considered a close relative. I’m not paying” (C.M. 33:2).

“Leave him out,” said Asher. “There are still two witnesses, Reuven and Shimon, who are not relatives. Isn’t that enough?”

“It is not,” answered Rabbi Dayan. “The Mishnah (Makkos 5b) teaches that if one witness in a group is found to be a relative or otherwise disqualified from testifying, the entire testimony is void. This is similar to the saying ‘One rotten apple spoils the whole bunch.’”

“Rebbi and Rav Yossi dispute whether this rule applies also to monetary issues,” continued Rabbi Dayan. “The accepted halachah is in accordance with Rebbi that it applies also to monetary testimony” (C.M. 36:1).

“Then how can anybody ever testify?” asked Asher. “In many instances relatives are present!”

“A relative disqualifies others only if he intended to serve as a witness with them,” replied Rabbi Dayan. “However, if he saw the event but had no intent to serve as a witness, he does not void the testimony of the other qualified witnesses. Furthermore, while the Rambam, Shulchan Aruch and Shach (36:8) maintain that the relative voids through intent alone, Tosafos and the Rema maintain that he voids the other witnesses only if he actually came and testified.”

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