Gemara

Question: R. Yochanan requires a majority regarding Rabbinic prohibitions!

Mishnah (Mikva'os 7:2): (The following sometimes disqualify a Mikveh, and sometimes help to complete the Shi'ur...fruit juice...and fermented Temed (water poured over grape pits and skins).) If a Mikveh had exactly 40 Se'ah, and a Se'ah was added to the Mikveh and a Se'ah was removed, it is Kosher.

R. Yochanan: This may be done up to a majority.

Suggestion: He requires that a majority (of Kosher water) remains!

Answer: No, he requires that a majority is not removed (but half may be removed).

Shabbos 15a - Mishnah - Hillel: A Hin (three Kavim) of Mayim She'uvim (water that was in a Keli) disqualifies a Mikvah;

Hillel said 'Hin', for one must repeat exactly what he heard from his Rebbi

Shamai says, nine Kavim is Posel;

Chachamim say, we accepted testimony in the name of Shemayah and Avtalyon that the Shi'ur is three Lugim.

Bava Basra 66b: Mayim She'uvim disqualifies mid'Rabanan.

Mikva'os 2:9 (Mishnah - R. Eliezer). If one put bottles on the roof to dry them and they became filled with water, if it is winter or if there is a little water in the pit, one may break the bottles (and the water will fall from the roof to the pit and make it a Kosher Mikveh);

R. Yehoshua says, in either case one may break or tilt them (so the water will fall from the roof to the pit), but he may not pour directly into the pit

Rishonim

Rif (Shevu'os 5b): Three Lugim of She'uvim disqualify an incomplete Mikveh. One may add She'uvim to a complete Mikveh, or to an incomplete Mikveh through Hamshachah.

Rambam (Hilchos Mikva'os 4:1,2): Mid'Oraisa one may Tovel in "Mikve Mayim" of 40 Sa'im, whether or not it is She'uvim.

Mid'Rabanan, Mayim She'uvim is Pasul for Tevilah. Furthermore, three Lugim of She'uvim disqualifies a Mikveh. Even though it is mid'Rabanan, it is learned from a Hekesh. "Ach Mayan u'Vor Mikve Mayim Yihyeh Tahor" - Mikveh is written between a spring, which is totally natural, and a pit, which is filled totally by man. A Mikveh may not be totally Mayim She'uvim like a pit, nor must it be totally natural like a spring.

If bottles put on the roof to dry out became filled with water, even if it is winter one may break the bottles or tilt them and the water inside is Kosher, for he did not fill the bottles himself. If he lifted them and poured the water the water inside is She'uvim.

Rebuttal (Ran Shevu'os 5b DH Malei): In the Mishnah (Mikva'os 2:9) the water in the bottles is considered She'uvim because he put the bottles out when there were clouds, or all the more so if it was raining. The 'little' water in the pit refers to 21 Sa'im, the majority of the Shi'ur; R. Eliezer holds like R. Eliezer ben Yakov, who permits Hamshachah to complete a Mikveh if the majority is Kosher. R. Yehoshua allows to fill an entire Mikveh through Hamshachah. R. Shimshon allows to totally fill a Mikveh with She'uvim through Hamshachah; it appears that the Rif agrees. They rule like R. Yehoshua because R. Eliezer is Shemuti. It was taught that the Halachah follows him only in four places (in all of Shas, and this is not one of them). Rav Acha of Shanza, the Ra'avad and the Ba'al ha'Ma'or rule like R. Eliezer ben Yakov, for the Halachah always follows him. It is best to be stringent.

Rosh (Mikva'os 1): R. Tam says that it is a tradition from Sinai that three Lugim of She'uvim disqualify a Mikveh at the beginning. If three Lugim of She'uvim fall into an incomplete Mikveh, they disqualify it mid'Rabanan. He learns from R. Eliezer's law, that if the first Revi'is of water in a Mikveh was She'uvim, it disqualifies it. If three Lugim of Mayim She'uvim fall into an incomplete Mikveh, they disqualify it. Surely, he is more stringent about Mayim She'uvim at the beginning because it is mid'Oraisa!

Rejection (Rosh citing Ri): No, he is more stringent about Mayim She'uvim at the beginning because it is a full Pasul Mikveh. A Revi'is is the Shi'ur Mikveh for Tevilas Kelim. We never find a tradition from Sinai regarding Mayim She'uvim. Even if we did, sometimes this is said about a mid'Rabanan law that is as clear as a tradition from Sinai.

Ra'avad (Ediyos 1:3): The Pesul of Mayim She'uvim is mid'Rabanan. Hillel's Rebbe'im said 'a Hin', as opposed to three Kavim (the Mishnah usually discusses Kavim) to show that Chachamim picked the largest (liquid) measure Shi'ur in the Torah to disqualify a Mikveh. Shamai says nine Kavim, for this is a Shi'ur (that helps in place of) for Tevilah (it suffices to shower the entire body, therefore Chachamim said that it permits a Ba'al Keri to learn Torah). Chachamim say three Lugim, for this is an important Shi'ur (it is the smallest Nesachim for Korbanos Tzibur).

Question: Mid'Oraisa, fruit juice is Pasul for a Mikveh. Why does the Gemara (Yevamos 82b) call this a Pesul mid'Rabanan?

Answer #1 (Tosfos Yevamos 82b DH Nosan): Mid'Oraisa, Tevilah of Kelim does not require 40 Sa'im (a Revi'is suffices). Granted, if the majority is fruit juice it is Pasul mid'Oraisa, but when half is water it is Kosher mid'Oraisa and Pasul mid'Rabanan.

Answer #2 Tosfos (ibid.): Mid'Oraisa, when each Se'ah of fruit juice is added it becomes Batel. There is no limit to how many times this may be repeated. Mid'Rabanan it is Pasul when the total amount of fruit juice that fell in is the majority.

Answer #3 (Tosfos ibid.): The Seifa of the Mishnah (Mikva'os 7:2) and R. Yochanan also discuss Temed, which is Pasul mid'Rabanan (like Mayim She'uvim).

Answer #4 (Riva, brought in Beis Yosef YD 201 DH veha'Ramban): The Pesul of fruit juice is only mid'Rabanan.

Poskim

Shulchan Aruch (YD 201:3): The 40 Sa'im of a Mikveh may not be She'uvim. If they are, it is Pasul.

Rema: If the entire Mikveh is She'uvim it is Pasul mid'Oraisa; we must be stringent about a Safek. If the majority is Kosher and a minority is She'uvim, it is Pasul mid'Rabanan, so we are lenient about a Safek.

Beis Yosef (DH veha'Ran): The Ra'avad holds that She'uvim is Pasul mid'Oraisa if one poured the water in. If by itself the water came from the Kelim to the Mikveh, it is Pasul mid'Rabanan. R. Shimshon holds that She'uvim in a Keli that is Mekabel Tum'ah is Pasul mid'Oraisa; if the Keli is not Mekabel Tum'ah, e.g. it is of stone, the Mikveh is Pasul mid'Rabanan.