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Is it brought down anywhere, something to the effect that if one realizes he was incorrect in a matter (can be a Torah-related matter, or a secular matter), he must retract his statement?

Preferably, I am looking for a source in Tanach, Mishna, or Gemara.

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The fifth chapter of Avos says one hallmark of a chacham, wise person, is that mode al haemes, he acknowledges the truth. Rashi says this refers to someone who has erred and acknowledges a correction.

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Yes. The Life Work of Shimon haAmsoni. Aggada Insight by Rabbi Yitzchak Blau

Cited in Talmud Bavli no less than four times, this braissa first appears in Pesachim (22b)See the Art Scroll page 22b3 notes 27 - 30

Shimon HaAmsuni- some say Nechemia HaAmsuni- was doresh all the ‘esem’ (the appearances of the word ‘es’) in the Torah. Upon reaching ‘Es Hashem Elokecha tirah,’ he was poresh (he divorced himself from his work). His disciples inquired: ‘Rebbe- what will be with all the esem that you have been doresh?’ To which he replied: ‘Just as I have received reward for my drisha, so too will I be compensated for my prisha.’ Until R’ Akiva came and was doresh that ‘Es Hashem Elokecha tirah’ is coming to include [the fear of] talmidei chachamim.

Some of the meforshim that I have seen explain that it was the retraction that allowed Rabbi Akiva to conclude that this es referred to Talmidei Chachomim. That a true talmid chacham would act in this way.

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