Gemara

29b - Beraisa: On Yom Tov, Ein Shonim (one may not sift again flour that was already sifted);

R. Papiyas and R. Yehudah ben Bava say, Shonim.

All permit if a pebble or chip fell in.

A reciter of Beraisos: Ein Shonim on Yom Tov. If a pebble or chip fell in, one may remove it by hand.

Objection (Ravina): It is forbidden, for this is like Borer!

Rava bar Rav Huna: Shonim flour on Yom Tov.

Rav Nachman: Tell Rava that we do not need his teaching. Sifters are used in Neharda'a (people know that it is permitted).

Rav Yosef's wife sifted using the back of the sifter (for a Shinuy). Rav Yosef told her that he wants to eat nice bread (she should sift normally)

Rav Ashi's wife sifted Al Gav a table (this will be explained).

Rav Ashi: My wife is the daughter of Rami bar Chama, who was meticulous about his deeds. Had she not seen this in her father's house, she would not do so.

Rishonim

Tosfos (DH a'Gava): Rashi explains that Al Gav a table is its back. Alternatively, it can mean on the table. This is a Shinuy, for usually one sifts into a bowl.

Rif and Rosh (3:16): On Yom Tov, Ein Shonim. R. Papiyas and R. Yehudah ben Bava say, Shonim. All permit if a pebble or chip fell in. Rav Yosef's wife sifted using the back of the sifter. Rav Yosef told her that he wants nice bread. Rav Ashi's wife sifted Al Gav a table. Rav Ashi said that she is Rami bar Chama's daughter, and Rami was often by Rav Papi, who was meticulous about his deeds. Surely she saw this done in her father's house!

Ran (DH Tanu): Rashi says that the first Tana forbids Shonim because it was possible from before Yom Tov. He holds that the Isur is mid'Rabanan, for mid'Oraisa we do not distinguish whether or not it was possible to do Melachah of Ochel Nefesh before Yom Tov. If a pebble fell in on Yom Tov it is permitted, for it was not possible before Yom Tov. Others say that the first Tana forbids, even though most people are Shoneh only for that day, for this can be confused with Meraked (sifting for the first time). Meraked is usually done at once for a long time, therefore it is forbidden. The other Tana'im do not decree. The Halachah follows the first Tana, who forbids. However, it is permitted through a Shinuy, like the cases brought. Perhaps no Shinuy is required according to letter of the law; we learn from Rav Yosef. Rav Ashi's wife was stringent.

Ran (DH v'Garsinan): R. Shmuel's text says 'many sifters are Meraked in Neharda'a', i.e. with a Shinuy; we learn from this. Rashi's text permits only Shonim, with a Shinuy.

Rosh (ibid.): The wives of Rav Yosef and Rav Ashi sifted for the second time. One may not sift the first time even through a Shinuy. The Rif brings both cases even though they seem to argue: Rav Yosef does not require a Shinuy, but Rav Ashi does. The Halachah should follow Rav Ashi (he is Basra). It seems that they do not argue. One need not use the back of the sifter (for a Shinuy), for this does not sift nicely. The Shinuy of the back of the table is required. This is unlike the Rambam, who equates them.

Rambam (Hilchos Yom Tov 1:7): Sifting is forbidden on Yom Tov, for it could be done beforehand.

Rambam (3:14): On Yom Tov, one may not sift flour a second time even if it was sifted before Yom Tov. It is permitted only if a pebble or chip fell in. With a Shinuy it is permitted, e.g. using the back of the sifter or on the back of a table or anything like this.

Ra'avad: The Halachah follows R. Papiyas and R. Yehudah ben Bava, who permit Shonim. If a pebble or chip fell in it is forbidden, but if Shinah, it is permitted.

Magid Mishnah: A Shinuy permits Shonim, but not Meraked. This is implied by the Rambam above (1:7). The Rambam learns from Rav Ashi's wife that a Shinuy is needed for Shonim; the Rif seems to agree. The Rashba and Ba'al ha'Itur do not require a Shinuy.

Question (Lechem Mishneh): A Shinuy permits Borer. Why doesn't a Shinuy permit Meraked? We cannot simply learn from the Rambam (1:7) that a Shinuy does not permit Meraked, for there he also forbids Borer, which is permitted through a Shinuy. Rather, since there he forbids (all) Meraked and here he permits only Shonim through a Shinuy, this shows that Meraked is always forbidden. The Magid Mishnah says that the Rif and Rambam agree. The Rif also brought Rav Yosef, who does not require a Shinuy. The Rosh explains that the Rif requires only a small Shinuy, but the Rambam disagrees, for he equates the back of the sifter and the back of a table. Perhaps the Magid Mishnah holds that the Rif and Rambam both require a small Shinuy (the back of a table), but not a big Shinuy (the back of the sifter).

Poskim

Shulchan Aruch (506:2): On Yom Tov, one may not sift flour for the first time, even through a Shinuy.

Mishnah Berurah (5): Melachos before kneading are forbidden. Some forbid them mid'Oraisa. Sifting is forbidden even if it was impossible before Yom Tov.

Shulchan Aruch (ibid.): If it was sifted before Yom Tov and a pebble or chip fell in, one may sift it again without a Shinuy.

Beis Yosef (DH Garsinan): The Magid Mishnah says that the Ra'avad learns from Rav Yosef to permit Shonim without a Shinuy. If a pebble fell in it is forbidden to remove it by hand, but one may be Shoneh (sift again). The Rashba permits removing it by hand; it seems that the Rambam agrees. The Kolbo cites the Ra'avad to allow Shonim if nothing fell in. If something fell it looks like Borer, and it is permitted only through a Shinuy.

Rebuttal (of Kolbo - Taz 3): The Beraisa says that we are more lenient when a pebble fell in. How could the Ra'avad disagree? Also, Borer is permitted on Yom Tov. Rather, we must explain like the Magid Mishnah. One may not remove a pebble through Borer because it is possible without an Av Melachah, through Shonim.

Question: Why did the Rif and Rosh omit Ravina's teaching, and why did the Rambam rule against it? Ravina is Basra, and no Amora argues with him!

Answers (Beis Yosef DH veha'Rif): Perhaps they understand that Ravina merely refuted what the reciter said. If you forbid Shonim, i.e. even through a Shinuy, all the more so you may not remove a pebble! However, we permit Shonim through a Shinuy, and removing a pebble is no more stringent. Alternatively, removing a pebble is not sifting; the Poskim relied on what they wrote regarding Borer. This seems correct.

Rebuttal (Taz 3): We do not find that they forbid this in Hilchos Borer! Rather, his first answer is correct.

Kaf ha'Chayim: Many say that the Shulchan Aruch does not distinguish, and permits even if the pebble fell in before Yom Tov. The Ran explicitly requires a Shinuy in this case, for one should have removed it before Yom Tov. It is proper to be stringent.

Mishnah Berurah (9): Some require a Shinuy even if it fell in on Yom Tov.

Rema: Some permit to remove the pebble by hand. Some forbid.

Mishnah Berurah (12) and Kaf ha'Chayim (14): It is proper to be stringent. If a fly fell into a cup on Shabbos, one may not remove it without taking some liquid as well. One should not drink an entire cup and leave only the dregs at the bottom. One may remove the pebble with some of the flour around it.

Shulchan Aruch (ibid.): If nothing fell in, but one wants to sift again to make nice bread, one may make a small Shinuy, e.g. on the back of a table.

Taz (3): Shonim is not an exertion. It is evident that it is not the first time, and it is unlike Meraked, so it is not Borer.

Magen Avraham (3): Our flour has no bran because it is already sifted during grinding, so perhaps we may sift it 'again' on Yom Tov.

Rebuttal (Mishnah Berurah 14): We may not sift it, for our sifters are finer and they hold back coarse flour that passed through the first time. This is like sifting for the first time!

Magen Avraham (4): It is normal to sift again at the time of (i.e. just before) kneading.

Mishnah Berurah (14): If one normally sifts onto a table, this is not considered a Shinuy. One may sift onto a cloth (Sha'ar ha'Tziyon 15).

Rema: Similarly, one may tell a Nochri to sift, even for the first time if he uses a Shinuy.

Source (Beis Yosef DH v'Ein citing Tosfos): We rely on R. Shmuel's text to tell a Nochri to be Meraked with a Shinuy.