Gemara

50b (Shmuel): If a knife was inserted between rows of bricks in a wall, it is permitted only if he removed and re-inserted it before Shabbos.

94b: A case occurred, and Rav Nachman bar Yitzchak permitted Hotza'ah of a Mes to a Karmelis.

R. Yochanan, brother of Mar brei d'Ravina: Is your ruling like R. Shimon? Granted, R. Shimon exempts from Chatas, but he agrees that it is forbidden mid'Rabanan!

Rav Nachman: It is even like R. Yehudah. I permitted only Karmelis, but not Reshus ha'Rabim. Kevod ha'Briyos overrides a Lav ("Lo Sasur", the basis of all mid'Rabanan laws).

100b: One may pour water on the side of the ship, and it descends to the sea by itself.

Question: Even though he does not pour it directly into the sea, his Ko'ach (impetus) causes it to go to the sea!

Answer: Chachamim did not forbid Ko'ach regarding Karmelis.

120b (Beraisa): If a lamp is in back of a door, one may open and close it normally. If the lamp extinguishes, this is no problem.

Rav cursed those who ruled like this, for R. Shimon agrees that a Pesik Reishei is forbidden.

146a (Rabah): Mid'Oraisa, one is not liable for a hole unless it is made for entrance and exit;

Chachamim decreed to forbid due to a hole in a chicken coop, which lets in air and lets out heat.

(Rav Nachman): The Halachah is, one may not widen a hole.

Eruvin 88a (Mishnah - R. Eliezer ben Yakov): If the first four Amos in Reshus ha'Rabim of a Biv (a drainage ditch) are covered, one may pour into it.

Chachamim forbid pouring into a Biv, even if it extends 100 Amos along a roof or Chatzer;

One may pour onto the roof, and the water will spill into the Biv.

Rishonim

Rosh (Pesachim 2:2): A Pesik Reishei is forbidden for all Torah laws.

Mordechai (Shabbos 375): May one pour water from Reshus ha'Yachid into a trough, and it flows into Reshus ha'Rabim? If it is not covered for four Amos in Reshus ha'Rabim, Maharam forbade. This is like Eruvin 88a. Seemingly, since we do not have Reshus ha'Rabim, and all our squares are like Karmelis, we should permit, for Chachamim did not decree about Ko'ach in Karmelis, like it says in Shabbos (100b) regarding a ship. This is wrong. The ship is on the sea or in a river. People will not come to confuse it with Reshus ha'Rabim. In the city, Chachamim decreed even about Karmelis, lest one confuse it with Reshus ha'Rabim, like we say in Eruvin 87b. A balcony above water requires a Mechitzah 10 Tefachim tall to permit drawing water. One opinion forbids spilling water through it. The other opinion permits even spilling. This is a mid'Rabanan law, so we follow this lenient opinion. Even so, here it is forbidden, for he intends for the water to leave.

Terumas ha'Deshen (64): One who removes and inserts a knife is not liable for increasing the hole, since it is not made for entrance and exit. We can permit one who does not intend, even if it is a Pesik Reishei, since it is mid'Rabanan. The Mordechai says so in the name of Maharam. However, the Mordechai must permit only a knife that was removed and re-inserted before Shabbos. It cannot be a Pesik Reishei to widen the hole, for one who makes a hole and does not need the earth is liable mid'Oraisa for Binyan. We say so about [removing] a shovel inserted in earth (Beitzah 8a).

Gra (on OC 314:1, DH v'Im): In Beitzah 8a, R. Shimon exempts one who digs and needs only the earth, due to Melachah she'Einah Tzerichah l'Gufah. Terumas ha'Deshen infers that if he does not need the earth, he is liable.

Damesek Eliezer: If he does not need even the earth, all the more so he is exempt! The Magen Avraham (318:36) said that this is not even considered a Pesik Reishei [to be liable for it].

Taz (2): There is a printing mistake in the Terumas ha'Deshen. It should say "if one digs and needs only the earth, it is forbidden only mid'Rabanan."

Mishbetzos Zahav (2): Based on the Taz' correction, the Terumas ha'Deshen concludes to forbid a Pesik Reishei even for a mid'Rabanan law, unlike his Hava Amina. The Magen Avraham (5) did not change the text. Also in the conclusion, the Terumas ha'Deshen permits a Pesik Reishei of a mid'Rabanan matter.

Note: The Magen Avraham notes that also Terumas ha'Deshen (66) permits a Pesik Reishei of a mid'Rabanan matter. It seems that he did not retract.

Tosfos (Shabbos 103a DH Lo): In Sukah (33b), R. Elazar b'Rebbi Shimon permits diminishing the berries on a myrtle branch, i.e. if he took it to eat. Even though it is a Pesik Reishei, a small improvement like this is forbidden only mid'Rabanan. Chachamim did not decree when he does not benefit, e.g. he already has Kosher branches.

Tosfos (Pesachim 25b DH Lo): Even R. Shimon forbids a Pesik Reishei [e.g. he drags a Kli, and benefits from the resulting furrow]. Do not say that in our Sugya he permits a Pesik Reishei, for dragging is only an Isur mid'Rabanan of plowing k'Le'acher Yad (unskillfully). We learn to Torah Isurim such as Me'ilah, Kil'ayim and Parah Adumah. Rather, he could avoid the Isur with difficulty, so it is not a Pesik Reishei.

Melei ha'Ro'im (on Tosfos): This connotes that a Pesik Reishei of a mid'Rabanan Isur is permitted, like the Terumas ha'Deshen. See the Magen Avraham (below).

Poskim

Shulchan Aruch (OC 314:1): If a knife was inserted in a Kli from before Shabbos, he may remove it and return it, for he does not intend to widen.

Magen Avraham (5): The Terumas ha'Deshen permits a Pesik Reishei for mid'Rabanan laws. We forbid a Pesik Reishei regarding mid'Rabanan laws (316:4). The Gemara clearly says so (Tosfos 103a). Also 337:1,2 connote like this. Even though Tosfos (25b) says that it is permitted, Tosfos means 'do not say...' Really, it is forbidden. Also 277:1,3 connotes like this. (We forbid a Pesik Reishei of extinguishing.) The Rosh forbids Pesik Reishei for all Torah laws. It seems that he said so to include other Isurim, but he forbids also for mid'Rabanan laws. There is no proof from the Mordechai. He did not do anything. Even if the water goes to Reshus ha'Rabim, it is not forbidden (357:1). Chachamim decreed in Reshus ha'Rabim, and (some texts - but) in Karmelis only when he intends to be Motzi. If he does not intend, he will not come to overtly be Motzi. One may tell a Nochri to do what is a Pesik Reishei, like I said in 253:41.

Gra (11 DH v'Im): It seems that the Terumas ha'Deshen retracted from what he wrote above to permit a Pesik Reishei for mid'Rabanan laws. Surely, [a knife in] a barrel is only mid'Rabanan.

Damesek Eliezer: In the conclusion, the Terumas ha'Deshen forbids inserting into a barrel due to a Pesik Reishei of a Torah law. If it were mid'Rabanan, he would permit without intent. However, the Gra explains that he concludes that one may not insert in the barrel in a way that is a Pesik Reishei. I.e. he forbids a Pesik Reishei of a mid'Rabanan law. We cannot learn [to permit] from a knife between bricks, for he merely separates them.

Gra (ibid.): His proof from Ko'ach [in Eruvin] is not a proof. Even in Reshus ha'Rabim there is no Isur. The Magen Avraham brought proofs against him. Shabbos 120b and Beitzah 36a prove that a Pesik Reishei is forbidden even for mid'Rabanan laws. We could reject this like the Ran said in the name of R. Chananel (94b), but we do not distinguish.

Damesek Eliezer: Regarding the ship, the person does nothing. It is permitted even if the water goes to Reshus ha'Rabim. The Gra's proof from 120b is because extinguishing is only mid'Rabanan, for it is Einah Tzerichah l'Gufah. Beitzah 36a forbids spreading a mat over a hive if it is a Pesik Reishei that bees will be trapped. Even with intent this is only mid'Rabanan, for bees are not a species that is trapped. On 94b, we asked that R. Shimon could forbid taking a Mes to Karmelis. Perhaps since R. Shimon exempts in Reshus ha'Rabim, he permits l'Chatchilah in Karmelis! The Ran answered that we say so for what is unlike a Torah Isur. For what is like a Torah Isur, but one is exempt due to Einah Tzerichah l'Gufah, Karmelis is like Reshus ha'Rabim. If so, we cannot learn from extinguishing and trapping to forbid a Pesik Reishei for a purely mid'Rabanan law! However, it is more reasonable not to distinguish.

Mishnah Berurah (11): The Magen Avraham, Eliyahu Rabah, Gra and R. Akiva Eiger forbid a Pesik Reishei d'Nicha Lei even for mid'Rabanan laws. However, the Magen Avraham and Gra say that even a knife in a wall should be permitted, for Melachah she'Einah Tzerichah l'Gufah is weaker than other Isurei mid'Rabanan, and it is also Mekalkel and k'Leacher Yad.

Kaf ha'Chayim (15): Even though some Acharonim permit Pesik Reishei d'Nicha Lei for mid'Rabanan matters, the Magen Avraham brought several proofs to forbid. We cannot prove the Mechaber's opinion from the Heter of an inserted knife. This could be due to Mekalkel. The Acharonim are stringent. One should be stringent, but one may tell a Nochri to do a Pesik Reishei.

Shulchan Aruch (337:1): Davar she'Eino Miskaven is permitted, unless it is a Pesik Reishei. Therefore, one may drag a big or small chair or bench, as long as he does not intend to make a furrow.

Rema (2): One may not brush clothes with a brush of bristles, lest he break the bristles.

Magen Avraham (5): Even though there is no Torah Isur, we forbid a Pesik Reishei.

Mishnah Berurah (316:5): If a wild bird flew into the house, one may not close the door or window, even though it is not trapped there. Chayei Adam permits when it is cold if he does not intend for the bird, even if it is a Pesik Reishei, since mid'Oraisa, trapping does not apply to it. One may not do so for other species, to which trapping applies mid'Oraisa.

Mishnah Berurah (16): If one chased out of a box all the flies he saw, he may close the box without checking lest there are others inside. He need not be so stringent, for this is a Safek Pesik Reishei about a mid'Rabanan law, since trapping does not apply to flies.

Mishnah Berurah (18): One may not spread a mat over a hive if it is a Pesik Reishei that bees will be trapped, even though trapping them is only mid'Rabanan.