Gemara

We are lenient about a Safek mid'Rabanan.

40a: A wilted turnip was brought to Rava's city [on Yom Tov]. He permitted to buy it, for surely it was not picked today.

Question: Perhaps it was brought from outside the Techum!

Answer: Something brought for one Yisrael is permitted for another.

45b - Question: Why do we permit rain? Perhaps it came from clouds outside the Techum!

Answer: This is a Safek mid'Rabanan, therefore we are lenient.

24a (Beraisa - R. Gamliel): Safek Muchan is permitted;

R. Yehoshua forbids.

(Shmuel): The Halachah follows R. Yehoshua.

24b (Rav Papa): The Halachah is, if a Nochri brought a gift to a Yisrael on Yom Tov, if some of that species are attached, it is forbidden until Motza'ei Shabbos bi'Chdei she'Ya'asu (until the time needed to detach them).

Shabbos 151b (Mishnah): If a Nochri brought flutes on Shabbos, one may not eulogize with them unless they came from a nearby place.

Question: What is the meaning of '[they came from] a nearby place'?

Answer #1 (Rav): [We know that] it came from nearby (inside the Techum);

Answer #2 (Shmuel): We are Chaishinan (this will be explained) that it came from outside the wall.

(Support (for Shmuel - Mishnah): If they made a coffin or dug a grave, a Yisrael may be buried in it.

Even though we do not know whether or not it was for a Yisrael, it is permitted mi'Safek. The same applies if we do not know if it came from outside the Techum!)

Note: Some texts delete this support for Shmuel.

A Mishnah supports Rav (some texts - Shmuel);

(Mishnah): If there is a bathhouse open on Shabbos in a city that is half Yisraelim and half Nochrim, on Motza'ei Shabbos one must wait the time needed to heat the water.

Rishonim

Rif (Shabbos 65a): The Halachah follows Shmuel, who is concerned lest they were brought from outside the Techum at night, even if we know that they were inside the Techum in the morning. One must wait on Motzei Shabbos until they could come from a near place. The Mishnah about the bathhouse supports Shmuel.

Rosh (Shabbos 23:9): Shmuel is lenient to assume that they came from outside the wall, and inside the Techum. Therefore, one may use them immediately on Motza'ei Shabbos. Also elsewhere, we find that "Chaishinan" means to be lenient. The Rif says that Shmuel explain that one may not use them 'unless they came from a close place', i.e. unless he waits until they could come from a close place. 'Perhaps they were outside the wall' connotes like Rashi. Also, how does the Mishnah support Shmuel, that we are stringent about a Safek? When half the city is Yisrael, it is not a Safek! Since there is no indication for whom it was warmed (i.e. there is no majority), it was warmed for all of them.

Ran (DH u'Matzinu): Rashi holds that whenever there is a Safek, we are lenient. Why does Rav Papa require bi'Chdei she'Ya'asu for a gift when some of that species are attached? R. Yonah answers that in such a case [it is not a Safek]. When there are attached Peros, people normally pick them on the same day [that they bring them]. Since most pick them today, we are not lenient. In Eruvin 40a, we said 'perhaps they are from outside the Techum.' This implies that we forbid a Safek! I answer for Rashi that we discuss a Safek to teach that since they were for another Yisrael, even Rav, who forbids a Safek, permits.

Poskim

Shulchan Aruch (OC 325:7): If it is a Safek whether Peros were picked for a Yisrael, or if it is known that they were, but it is a Safek whether they were picked today, they are forbidden during the day and permitted at night bi'Chdai she'Ya'aseh.

Beis Yosef (DH v'Im): We forbid Safek Muchan. Semag, Sefer ha'Terumah and Hagahos Maimoniyos permit immediately at night, because it is a Safek. This is because we hold like Shmuel. They explain, like Rashi, that we are lenient to assume that it came from nearby. Therefore, one may use them immediately on Motza'ei Shabbos. Why did the Tur omit the Rif's opinion, that one must wait on Motza'ei Shabbos until they could be brought from elsewhere? Perhaps he relies on later in the Siman, where he brought the argument of Rashi and the Rif, and said that the Rosh rules like Rashi.

Gra (DH Safek): The Rif learns from Eruvin 40a. It says 'perhaps it was brought from outside the Techum', i.e. it is a Safek. If it was not brought for Nochrim, it is forbidden due to Safek.

Damesek Eliezer (16): According to Rashi's text, Tana'im argue like Rav and Shmuel, so we rely on the rule that the Halachah follows Rav in Isurim. Tosfos (151a DH Deika) quotes a text in which we bring a support only for Shmuel, so the Halachah follows him.

Mishnah Berurah (30): The Safek is when the Nochri told the Yisrael 'I will give to you Peros on Shabbos', and he went to the Nochri's house. He does not know whether the Nochri picked these for him, or intended to pick others for him, and picked these for himself. If the Nochri gave a gift to the Yisrael, it is not a Safek. Surely he picked for him, or if it is warm bread, he baked it for him (Magen Avraham, Tosefes Shabbos and other Acharonim, unlike the Bach).

Kaf ha'Chayim (63): The Bach holds that even a gift is a Safek whether he picked them for him.

Mishnah Berurah (31): Also if he brought Peros to sell, surely he picked them on Shabbos. Presumably, he brought the best, which was picked today. All forbid until Motza'ei Shabbos bi'Chdei she'Ya'asu.

Shulchan Aruch (ibid.): Some permit immediately at night.

Taz (5): It seems that also the Rosh holds like this.

Magen Avraham (14): We are not stringent to wait bi'Chdei she'Ya'asu due to Safek. Sefer ha'Terumah explains that amidst Safek, we are Choshesh [assume] leniently. Why in Sa'if 16 did the Mechaber write Stam that we are stringent? Perhaps we are more stringent about a Mes.

Gra (DH v'Yesh): In 498:3,4, the Mechaber wrote Stam like the Rashba, who is always lenient about a Safek mid'Rabanan. Surely it is forbidden on the same day, for it is Safek Muchan (Sa'if 5). The first opinion holds that the Sugya in Eruvin 45b holds like Rav in Shabbos 151a, but we hold like Shmuel.

Damesek Eliezer (17): In Sa'ifim 9 and 15, the Mechaber forbids something that is Safek from outside the Techum, like the Rif. The Magen Avraham (20) says that in Sa'if 9, the Mechaber forbids on Shabbos itself, and permits immediately after Shabbos, like v'Yesh Omrim (the lenient opinion) here, who forbid [only] on Shabbos. The Gra disagrees, and he is correct. If it is a Safek that it was picked or trapped today, this is Safek Muchan. We can say that it is forbidden on Shabbos, but not after Shabbos. However, if it came from outside the Techum, v'Yesh Omrim permit a Safek. Surely it is permitted even on Shabbos! The Beis Yosef explicitly says so according to Rashi. He said that the text of the Tur should say 'Rashi', and not 'Ri'. I say that the text is correct. The Tur refers to the Ri in Tosfos 151a, whose text omits 'our Mishnah is like Rav.' If so, the Halachah follows Shmuel, who is lenient.

Mishnah Berurah (33): This opinion agrees that it is forbidden on Shabbos itself. The Eliyahu Rabah says that one may rely on this opinion. One need not be stringent on Motza'ei Shabbos. The Chayei Adam says to be lenient only for the sake of a Mitzvah. Eshel Avraham (18) connotes like this, and so decided the Beis Yosef (515).

Kaf ha'Chayim (67): When the Mechaber is stringent Stam, and brings that some are lenient, one may rely on the lenient opinion in pressed circumstances, or for the sake of a Mitzvah.

Shulchan Aruch (9): If it is a Safek whether they were brought from outside the Techum, they are forbidden, i.e. if the Nochri lives outside the city. Even if the Nochri has one house in the Techum and one outside, we are lenient, even for the one for whom he brought.

Shulchan Aruch (15): If a Nochri brought flutes for a Yisrael to eulogize with on Shabbos, no Yisrael may eulogize with them until the time to bring them from nearby on Motza'ei Shabbos. If they were not brought through Reshus ha'Rabim, rather, through Karmelis, since no Isur Torah was done, one may use them immediately on Motza'ei Shabbos.

Shulchan Aruch (16): If it is a Safek if they were brought from inside or outside the Techum, we are concerned lest they were from outside.