Gemara

38b (Rav Yehudah): One who frees a slave transgresses an Aseh - "forever you will make them work for you."

Question (Beraisa): A case occurred in which R. Eliezer found only nine men in a Beis ha'Keneses, and he freed his slave to complete the Minyan.

Answer: Rav Yehudah agrees that one may free a slave if needed for a Mitzvah.

38a: Men were sinning with a Shifchah in Pumbadisa.

Abaye: If not for Rav Yehudah's law, I would force her master to free her.

Ravina: Rav Yehudah agrees that one may free a slave to prevent transgression!

Question: Surely, Abaye agrees! R. Chinena bar Rav Ketina taught that a case occurred in which Chachamim forced a master to free his half-Shifchah. Rav Nachman bar Yitzchak explained that men were freely sinning with her.

Answer: One may free a half-Shifchah, for there is no other way to stop transgression. She may not marry a slave, nor a free men. One can pair a Shifchah with a male slave; he will stop others from sinning with her!

41a (Mishnah - Beis Shamai): A half-slave cannot marry a Shifchah, nor a Bas Yisrael. "Lasheves Yetzarah" (the world was created for procreation)! Rather, for Tikun ha'Olam, we force his master to free him, and the half-slave writes a document obligating himself to pay half his value to his master.

Beis Hillel retracted, and agreed to Beis Shamai.

Shabbos 4a (Rav Bivi bar Abaye) Question: If Reuven stuck a dough in an oven on Shabbos, may he remove it before it bakes, to avoid a Chiyuv Chatas?

(Rav Shila): Rav Bivi really asked whether or not someone else (who knows that Reuven put it in b'Shogeg) may remove it to exempt him.

Objection: One may not sin (even mid'Rabanan) to save someone else from sin!

Rishonim

Rif and Rosh (4:23): The Halachah follows Ravina, who says that we free even a full Shifchah if she is sinning with men.

Rambam (Hilchos Avadim 9:6) One may free a slave for a Mitzvah, even a Mitzvah mid'Rabanan, e.g. to complete a Minyan or similar cases. If men treat a Shifchah like Hefker (for Bi'ah), she is a Michshol (stumbling block) for sinners. We force her master to free her, so she will marry and remove the Michshol.

Rosh: We permit freeing a slave also for a Mitzvah of the Rabim.

Tosfos (41b DH Kofin): One may not do even a light sin to save someone else who transgressed from a bigger sin. They forced a master to free his half-Shifchah because she made herself available for Zenus, and it is as if men were coerced. Also, we are more lenient about a Mitzvah of the Rabim.

Poskim

Shulchan Aruch (YD 267:79): One may free a slave for a Mitzvah, even a Mitzvah mid'Rabanan, e.g. to complete a Minyan. If men treat a Shifchah like Hefker, we force her master to free her, so she will marry and remove the Michshol.

Shevus Yakov (1:16): A Chacham permitted a community leader to leave Techum Shabbos in order to prevent someone from being Moser (inciting the kingdom to seize money of) great Tzadikim. He erred. Surely, one may not desecrate Shabbos to save money. We do not do even a light Isur to save one who did improperly. Perhaps the Chacham relied on Nachalas Shiv'ah (83), who relied on his reasoning to rule unlike Tosfos, the Rashba and the Shulchan Aruch. We desecrate Shabbos for a Choleh, in order that he will keep may Shabbosos. Surely, he sinned; death does not come without Chet. If we save from death, all the more so we save from (sin, which causes) eternal death! This is all wrong. Death does not come without Chet, i.e. Shogeg. Here, the Moser was Mezid. Even though making someone sin is worse than killing him, we transgress (almost) all Aveiros when necessary for life. Surely we do not sin for one who intentionally forfeits his eternal life!

Note: The Shevus Yakov connotes that we would not desecrate Shabbos to save a Choleh who sinned b'Mezid (this includes unhealthy or risky behavior - "v'Nishmartem l'Nafshoseichem"). Chachmas Shlomo (OC 329:1) says similarly. However, Divrei Yatziv (OC 167:1) explains that Chachmas Shlomo forbids Piku'ach Nefesh only for one who put himself is mortal danger. Igros Moshe (OC 1:127 DH Mah) permits Piku'ach Nefesh in any case.

Taz (YD 334:1): Mahari Mintz (5) permitted a man to marry Meinekes Chavero (a woman nursing another man's baby. Chachamim forbade her and Me'uberes Chavero (one pregnant from another man) to marry, lest she become pregnant and wean the baby prematurely, and he will die), to save them from going to Tarbus Ra'ah. He learned from Gitin 38.

Chasam Sofer (3 EH 36 DH Al): A case occurred in which a licentious single woman was blatantly pregnant, and a lewd man wanted to marry her. She threatened to give the baby to an orphanage if Rabanan would not allow them to marry, and both of them might sin greatly. We may not do even a light sin to save her. Surely, if one says that he will abandon Torah if we do not permit an Isur, we do not permit even an Isur mid'Rabanan for him. Mahari Mintz permitted only because we are more lenient about a Zonah. However, in our case we permit because the Isur of Me'uberes Chavero was due to concerned for the baby. Here, it is better that the baby be in mortal danger than to be put in a place where he will become an idolater. Making someone sin is worse than killing him! In the case of Mahari Mintz, had he been stringent the child would have grown up among harlots, but at least he would believe in Torah. Therefore, he needed to find another Heter (Zonah). Also, there she was Mezanah only once, but here she caused many men to sin.

Ein Yitzchak (1 EH 1:7): Since Yisre'elim are Arevim (responsible) for each other, why do we ask if one may sin to help another? He helps himself! Perhaps it is because Tosfos (Shevu'os 39b DH Hasam) says that the punishment for not protesting is not so great. Also, the punishment due to Arvus is borne by everyone, so an individual's share is small. If one sins himself, he bears the primary punishment.

Mishneh Halachos (11:532): We do not tell one to sin to help another, so if a daughter returns home late, a man should not wait for her (to light Neros Chanukah). On the other hand, if she will not light by herself or see the Neros, she will get used to neglecting the Mitzvah. In several places, Tosfos says that we permit one to sin in order that his children merit. One is responsible for his children more than mere Arvus. One who does not protest against his household is punished for their sins. Therefore, he should wait for her, since the Rema says that nowadays we publicize the miracle only to the household (so it is not so important to light immediately).