Gemara

(Beraisa - R. Shimon ben Gamliel): A woman may be Toleh (attribute blood found) to a Shomeres Yom k'Neged Yom on the day after she saw;

Rebbi says, she may not be Toleh. Therefore, both women are in Safek.

(Rav Chisda): They would argue also about a Tahor and a Tamei who walked on two paths. (One path has Tum'ah underneath that is Metamei anyone who traverses the path. We do not know which path each traversed.)

Objection (Rav Ada): Rebbi is Metamei both women only when they are similar. Here, the Tamei does not lose if we are Toleh that he walked on the Tamei path!

Answer: Rav Chisda holds that also the women are not so similar. The Shomeres Yom is not Tehorah until she immerses.

(R. Yosi b'Rebbi Chanina): If a Tahor and one who was Tamei or Taluy (Safek Tamei) walked on the two paths, all agree that we are Toleh that the Tahor walked on the Tahor path, and the Tamei or Taluy on the Tamei path.

Kerisus 17b (Rav): If Shumen and Chelev were in front of him, and he does not know which he ate, he is liable. If there was one piece and he ate it, he is exempt.

Question: What is his reason?

Answer #1 (Rava): He learns from "v'Asesah Achas mi'Kol Mitzvos Hash-m" He must be Shogeg about two Mitzvos (matters forbidden due to Safek).

Answer #2 (R. Zeira): When there were two pieces, it is possible to resolve the Safek (now). If there was only one piece, we cannot resolve it.

18a - Answer #3 (Rav Nachman): When there are two pieces, the Isur was Nikva. (There definitely was Isur there.) When there was only one piece, the Isur was not Nikva.

Kidushin 5b: If he gave the (Kidushin) money and she said the words, it is a Safek. Mid'Rabanan we are stringent and consider her Safek Mekudeshes.

39a (Rav Asi): A tradition from Moshe from Sinai forbids Orlah in Chutz La'aretz.

Question (R. Zeira - Mishnah): In Surya, Safek Orlah is permitted. (If Orlah of Chutz la'Aretz is forbidden mid'Oraisa, we must be stringent about a Safek!)

Answer (Rav Asi): The tradition permits Safek and forbids Vadai.

73a (Mishnah): Mid'Oraisa, a Shetuki (one who recognizes his mother but not his father) is Kosher. It says "Lo Yavo Mamzer (bi'Khal Hash-m)." A definite Mamzer is forbidden, but a Safek Mamzer is permitted;

A Mamzer may not marry a Vadai member of Kehal Hash-m (someone of proper lineage), but he may marry a Safek.

Yoma 74a (Beraisa): "Kol Chelev" forbids Chelev of a Kvi (a Safek Chayah).

Surely, this is an Asmachta. A verse would not include a Safek!

Rejection: The Tana holds that it is a species unto itself.

Sotah 29a (Rav Gidal) Contradiction: "If meat (of Korbanos) touches anything Tamei, it will not be eaten" implies that Vadai Tamei is forbidden, but one may eat Safek Tamei;

"Every Tahor person may eat meat (of Korbanos)" implies that a Vadai Tahor person may eat, but a Safek may not eat.

Resolution: When an involved party has understanding, we are stringent about a Safek. When no involved party has understanding, we are lenient.

Chulin 42a: "Zos ha'Chayah Asher Tochal" implies that there is another Chayah (living animal) that you may not eat, i.e. a Terefah.

Rishonim

Rif and Rosh (Kidushin 2a and 1:1): If he gave the Kidushin money and she said the words, we are concerned mid'Rabanan, and she needs a Get.

Rambam (Hilchos Ishus 3:2): If he gave the money and she said the words, she is Mekudeshes mi'Safek.

Ran (Kidushin 2a DH Tanu): The Rambam does not say 'mid'Rabanan'. Perhaps no Safek Kidushin requires a Get mid'Oraisa, for we follow the Chazakah (she is single). Mid'Rabanan, we are stringent about Safek Ervah.

Pri Chodosh (YD 110 Klalei Sefek-Sefeka 1, the last DH v'Im): Why does the Ran imply that the Rambam holds that the Safek is mid'Oraisa? The Rambam holds that every Safek is mid'Rabanan! It seems that the Ran was unsure about a Safek about how we rule. Then, perhaps even the Rambam requires a Get mid'Oraisa.

Rambam (Hilchos Isurei Bi'ah 18:17): The Isur of every Safek is mid'Rabanan. Therefore, Chachamim were lenient about a woman taken captive.

Rambam (Hilchos Tum'as Mes 9:11): If a Nefel (miscarriage) is in a pit in which weasels are common, and we do not know whether or not the Nefel is Metamei (and someone towered over it), we are lenient about the Safek.

Rambam (12): All these and similar Tum'os are mid'Rabanan. Mid'Oraisa, only Vadai Tum'ah is Metamei. Every Safek about Tum'ah, forbidden food, Ervah and Shabbos is mid'Rabanan. Even so, a Safek about a Chiyuv Kares is Asur mid'Oraisa, for one brings an Asham Taluy for it.

Maharit (DH v'Tzarich): The Rambam (Avos ha'Tum'ah 16:1) says that we are lenient about Safek Tum'ah in Reshus ha'Rabim, since Korban Pesach overrides even Vadai Tum'ah, and all the more so Safek Tum'ah, which is only mid'Rabanan. The Yerushalmi learns from Sotah that Safek Tum'ah in Reshus ha'Rabim is Tahor, but not because Sefekos are mid'Rabanan! Also the Bavli learns Safek Tum'ah in Reshus ha'Rabim from Sotah! It seems that we did not need to learn from Sotah. However, we need to distinguish Reshus ha'Rabim from Reshus ha'Yachid, so we will not learn from all Safek Tum'ah from Sotah. The Yerushalmi shows that we distinguish an individual from the Rabim. If Sefekos were not mid'Rabanan, we could not distinguish like this.

Note: Korban Pesach may be brought in Tum'ah only if the majority of Yisrael (or Kohanim or Kli Shares) is Tamei, but three make a Reshus ha'Rabim for which we are lenient about Safek Tum'ah!

Maharit (DH u'Mah): Even if the Torah is lenient about a Safek, it must teach that we follow the majority for when there is Chezkas Isur, and to teach that we follow the majority even to be stringent.

Maharit (DH Od): R. Shmuel Chaivan proves that the Rambam is lenient even when there was Chezkas Isur. If Kil'ayim was lost in a garment, we dye it and look for threads that accepted the dye differently (like wool and linen do). If we do not find this, we assume that the Asur threads fell out. I say that this is unlike two pieces, one of which is forbidden. It is like when there were two pieces, and a Nochri ate one and a Yisrael ate the other. Rebbi obligates an Asham Taluy, and Chachamim exempt. Chachamim require that the Isur is Vadai around when he eats. The Rambam rules like Rebbi. However, Kil'ayim is different, for the dying indicates that the Kil'ayim fell out. If not, the Safek (that it fell out) would not override the Vadai (that there was Kil'ayim).

Poskim

Pri Megadim (YD 110, Dinei Sefek-Sefeka 25): The Rambam agrees that Safek mid'Oraisa l'Chumra in eight cases:

There is Chezkas Isur. We stone based on Chazakos (Kidushin 80a)!

It is possible to clarify. We learn from Korban Pesach.

The Isur was Nikva, e.g. there were two pieces, one of which is Chelev.

It is a Safek Isur Kares (according to some texts).

Trei v'Trei (there are contradictory pairs of witnesses), or Chachamim did not resolve the Halachah. Trei v'Trei is a Safek mid'Rabanan only when there is a Chazakah.

The Isur is more common than the Heter.

A Safek in which in any case he does an Isur (e.g. he did Melachah all of Bein ha'Shemashos on Erev Shabbos and Motza'ei Shabbos - Hagahos in Friedman Shulchan Aruch).

Safek Tamei is like Vadai in Reshus ha'Yachid, and Tahor even mid'Rabanan in Reshus ha'Rabim.

Noda bi'Yehudah (2 YD 38): Even though the Rambam gave a general rule, and said that only mid'Rabanan we are stringent about Sefekos about Tum'ah, Ervah, Shabbos..., a Safek adulteress is called Safek Sotah, and the Torah made this Safek like Vadai (Isur).

Noda bi'Yehudah (EH 1:66): Tosfos in Kesuvos (26b DH Anan) and Kidushin (66a DH Mai) forbid a Shevuyah (a woman who was taken captive) to a Kohen mid'Oraisa. Tosfos Kidushin 12b (DH Im) connotes that it is only mid'Rabanan. Really, it is a Safek mid'Oraisa. The Rambam holds that all Sefekos are mid'Rabanan, but other Poskim forbid mid'Oraisa. A Shevuyah lost her Chazakah, for most Nochrim are promiscuous. Chachamim were lenient because she makes herself repulsive. Kesuvos 13b says that (it is as if) there are witnesses about a Shevuyah (that she was defiled). Without a decree, it is a Safek Isur Torah. Chachamim decreed to make it like Vadai, like the Shach (YD 110 Klalei Sefek-Sefeka 17,18) says about cheese of Nochrim. No Gemara explicitly says that Chachamim decreed. Perhaps mid'Oraisa we rely on Chazakah, and Chachamim decreed to make it a Safek. If so, they decreed only for lineage, about which we are stringent, but not for fines or Kesuvah. Tosfos says that we decreed about a fine, lest a Kohen marry her.

Suggestion: If mid'Oraisa it is a Safek, even without a decree she does not receive a fine!

Rejection (Noda bi'Yehudah): Due to the decree, if she seized the money she must return it. However, we can say that seizure of a fine never helps.

Chidushei ha'Rim (Chulin 42a): The Gemara expounded that a Terefah does not live. Granted, if the Torah forbids Sefekos, we learn from this that if a Safek Terefah lived 12 months, it is Kosher. However, if the Torah permits Sefekos, in any case it is permitted! According to the Poskim that the Torah is stringent when it is possible to clarify, like it says in Kerisus, the verse teaches that we must be stringent in this case and wait 12 months! According to the Poskim that the Torah is lenient even when it is possible to clarify, we can say that a Vadai Terefah became mixed with one Kosher animal. Since the Isur was established, the Torah is stringent about the Safek. However, this is difficult, because the Gemara (Zevachim 74b) had difficulty finding a case of a Terefah that became mixed. (Almost any Terefah that was once known is still recognizable.) It seems that the Rambam learned that the Torah is lenient about a Safek from Sotah 29a. The Torah forbids (Kodshim) meat that is Vadai Tamei, but forbids a Safek Tamei person to eat. This shows that when the Torah forbade matters such as Chelev, it forbade Vadai Isur, but not a Safek. Therefore, the Rambam says that the Torah permits a Safek. However, when the Torah permits, it permits a Vadai. Therefore, the Torah forbids a Vadai Terefah, but a Safek Terefah is permitted. However, "this is the Chayah that you will eat" permits only Vadai Chayah (what will surely live, i.e. it is not Terefah). Therefore, the Torah needed to teach that if it lives 12 months, it is Kosher. Really, to answer the question we can say that "Zos ha'Chayah" was not extra to teach that a Terefah lives. It is relevant to us, for we forbid a Safek Torah mid'Rabanan. This is why the Gemara called it a hint. Alternatively, it is a case in which the majority of times it is Terefah. The Torah forbids, but it is permitted if it lives 12 months.

Merumei Sadeh (Kidushin 73a DH v'Ra'uy): The Rambam learns from Toras Kohanim that we must be stringent about a Safek that can be clarified. "Bein ha'Chayah ha'Ne'echeles..." teaches that we must clarify whether an animal is Tereifah or Kosher. Also, we may cut "only a tree that you know that it is not a fruit tree", i.e. we must check. The tradition about Orlah in Chutz la'Aretz teaches that we need not check. Likewise, in Chulin (9a) the Gemara derived that we rely on Chazakah or a majority to permit, i.e. without checking.