Gemara

(Mishnah - Beis Shamai): Hechsher Mavoy (what is necessary to permit carrying in an aleyway) is a Lechi and a Korah;

Beis Hillel say, it is a Lechi or a Korah.

Inference: Beis Shamai require a Lechi and a Korah [to make a proper Mechitzah] because they hold that mid'Oraisa, Reshus ha'Yachid needs four walls!

Rejection: They hold that three walls [make Reshus ha'Yachid mid'Oraisa, and] are Mechayev one who throws to there [from Reshus ha'Rabim]; they require four walls to carry [mid'Rabanan].

Inference: Beis Hillel require only a Lechi or a Korah, because they hold that mid'Oraisa, three walls makes Reshus ha'Yachid!

Rejection: They require only two walls regarding [Chiyuv for] throwing. They require three [proper] walls [mid'Rabanan] to carry.

12b (Rav Yehudah): If a Mavoy was Huchshar through a Lechi, one who throws to it is liable;

If it was Huchshar through a Korah, one who throws to it is exempt.

Inference: He holds that a Lechi is considered a Mechitzah, but a Korah is only a Heker (reminder). Rabah explicitly said so.

(Rava): Both a Lechi and a Korah are only Hekerim.

Question (R. Yakov bar Aba - Beraisa): If one throws to a Mavoy, he is liable only if it has a Lechi.

Answer (Rava): It means, if it needs only a Lechi, he is liable. If it needs more than a Lechi (it is Mefulash), he is exempt.

Question (Beraisa - R. Yehudah): If one owns two houses on opposite sides of Reshus ha'Rabim, he puts a Lechi on each side [of one house], or a Korah on each side, and he may carry in the middle.

Chachamim: We cannot be Me'arev Reshus ha'Rabim like this!

Answer: R. Yehudah holds that two walls make a Reshus ha'Yachid mid'Oraisa.

94a (Mishnah - R. Eliezer): If [the wall of] a Chatzer was breached to Reshus ha'Rabim, one who throws from it to Reshus ha'Yachid or vice-versa is liable;

Chachamim exempt, for it is a Karmelis.

Rishonim

Rambam (Hilchos Shabbos 14:4): A corner next to Reshus ha'Rabim surrounded by three walls, and the fourth side is [open] to Reshus ha'Rabim, e.g. a Mavoy without a Lechi or Korah, is a Karmelis.

Rambam (17:9): A Mavoy that became permitted through a Korah, even though one may carry in all of it, like Reshus ha'Yachid, one who throws to it from Reshus ha'Rabim or vice-versa is exempt, since a Korah is a Heker.

Rebuttal (Ra'avad): Since one may carry in all of it, it has three walls. We hold that three walls [makes Reshus ha'Yachid] mid'Oraisa! If so, one who throws into it is liable. Did Chachamim come to be more lenient than Torah law to obligate a Korah?!

Rambam (ibid.): If it became permitted through a Lechi, one who throws to it from Reshus ha'Rabim or vice-versa is liable, since a Lechi is like a Mechitzah on the fourth wall.

Magid Mishneh: On 11b, the text of the Rambam and R. Chananel says 'perhaps Beis Shamai hold that three walls forbids', unlike our text, which says 'is Mechayev.' On 94a, when a Chatzer was breached on the fourth side, Chachamim exempt one who throws from it to Reshus ha'Rabim, for it is a Karmelis. The simple reading is like the Rambam. However, the Rambam's Perush in the first Perek is difficult. The Ra'avad, Tosfos and the Rashba hold that three walls make Reshus ha'Yachid mid'Oraisa, and they had difficulty explaining 94a. Some distinguish between whether it is open to Reshus ha'Rabim or Karmelis.

Ramach (in Kesef Mishneh): Beis Hillel obligate due to three walls! If the Rambam will say that when it is open to Reshus ha'Rabim, it is Karmelis even if it has three walls, it should be forbidden to carry inside. Chachamim enacted Eruvin to be stringent. A loaf does not make Reshus ha'Yachid (12b)!

Note: The Rambam holds that mid'Oraisa, it is Mekom Petur. Chachamim were stringent to require Eruvin to permit carrying. Rav Sheshes asked that a loaf does not make Reshus ha'Yachid to obligate one who throws into it!

Ramach (ibid.): We must distinguish what opens to Reshus ha'Rabim from what opens to Karmelis. On 94a, when one wall fell, we exempt one who throws to Reshus ha'Rabim. In the first Perek, we obligate one who throws into a place with two walls [and a Lechi - PF]. This shows that two walls make a Reshus ha'Yachid.

Rosh (Sukah 1:9): Rav Yehudah holds that if a Mavoy was Huchshar through a Lechi on one side, one who throws to it is liable. I.e. it was open on two sides. He holds that a Lechi is considered a Mechitzah.

Rashi (12b DH Eino): If a Mavoy needs only a Lechi, i.e. it is not Mefulash (it has three walls), one who throws to it is liable.

Tosfos (12b DH Eisivei): Even Rabanan [who say 'one cannot be Me'arev Reshus ha'Rabim like this'] require three walls only mid'Rabanan.

Poskim

Shulchan Aruch (OC 345:2): Reshus ha'Yachid is a place surrounded by walls 10 Tefachim tall and it has at least four by four Tefachim...

Beis Yosef (DH Reshus): The Beraisa calls it a proper Reshus ha'Rabim to teach that it has the proper number of walls, i.e. it is surrounded on every side. Regarding a ditch, we say Gud Asik (we project the walls upward), and it is surrounded on all four sides.

Suggestion: The Tur did not specify how many walls are required to be Reshus ha'Yachid, to avoid entering the argument. The Rambam requires four walls, or three and a Lechi on the fourth side, for we hold that a Lechi is considered a Mechitzah. The other Poskim hold that three walls suffices to obligate one who throws into it from Reshus ha'Rabim, like the Magid Mishneh explains.

Rejection (Beis Yosef, ibid.): All agree that three walls do not make it a full Reshus ha'Yachid to permit carrying inside even mid'Rabanan. Since he discusses a full Reshus ha'Yachid, he should have taught that three walls do not suffice! Also, later he explicitly wrote the Rambam's opinion, that a Mavoy with three walls without a Lechi or Korah is Karmelis. We must say that he did not teach that four walls are required, for this would connote full walls, and that a Lechi or Korah does not help. He relied on what he explains below.

Magen Avraham (1): The Poskim connote that two walls and a Lechi make Reshus ha'Yachid mid'Oraisa, for a Lechi is considered a Mechitzah. The Rosh says so. See 363:11. The Rambam holds that three walls makes a Karmelis. Four L-shapes, one in each corner, like Pasei Bira'os (which are made around wells to permit drawing water) make Reshus ha'Yachid mid'Oraisa (Eruvin 20b).

Shulchan Aruch (14): A corner next to Reshus ha'Rabim (Rema - e.g. a Mavoy with three walls, and no Lechi or Korah on the fourth side) is a Karmelis.

Beis Yosef (DH u'Mah): The Tur says so. This is like the Rambam. A Lechi would make it a full Reshus ha'Yachid, and a Korah would permit carrying inside.

Mishnah Berurah (55): Even if it is Huchshar through a Korah, it is a Reshus ha'Yachid regarding carrying inside. This is like the Rambam. The other Poskim hold that three walls makes Reshus ha'Yachid mid'Oraisa, but carrying is forbidden mid'Rabanan, like a Karmelis.

Shulchan Aruch (363:1): A place with three Mechitzos, Chachamim forbade carrying there until making a Tikun on the fourth side.

Beis Yosef (DH Min): The Tur says that mid'Oraisa, three walls makes a full Reshus ha'Yachid. If this means that one who throws from it to Reshus ha'Rabim is liable, this is unlike he said in Siman 345, that a Mavoy with three walls is a Karmelis! Rather, he means that it is a full Reshus ha'Yachid to permit carrying inside. Others say that one who throws from it to Reshus ha'Rabim is liable.

Rebuttal (Taz 1): It is difficult to say that the Tur said 'full Reshus ha'Yachid' merely to permit carrying. Rather, in Siman 345 he discusses 'a corner next to Reshus ha'Rabim.' When it is crowded, sometimes people of Reshus ha'Rabim go there (Shabbos 7a). The place itself is a full Reshus ha'Yachid, but Reshus ha'Rabim is Mevatel it from being Reshus ha'Yachid. It is Batel to Reshus ha'Rabim, and is called Tzidei Reshus ha'Rabim. It is like a Karmelis. We do not discuss such a case here, so it remains a full Reshus ha'Yachid. Even though he says 'whenever there are three walls...' he means when there is no Bitul due to the Rabim going there when it is crowded. The Magid Mishneh says that some distinguish between whether it is open to Reshus ha'Rabim or Karmelis. The Tur says 'whenever' to include what is open to Reshus ha'Rabim, when the Rabim do not go there when it is crowded. Alternatively, 'whenever' teaches a Chatzer and Mavoy.

Kaf ha'Chayim (1): The Bach, Magen Avraham and others say that the Tur holds that in every case, three walls is a full Reshus ha'Yachid mid'Oraisa.

Bi'ur Halachah (DH Asru): The Rambam and R. Chananel hold that a full Reshus ha'Yachid mid'Oraisa requires four walls, or three walls and a Lechi. Most Poskim hold that three walls, or two walls and a Lechi, make a full Reshus ha'Yachid mid'Oraisa. Eruvin 12b proves that it does not matter whether the two [full] walls are adjacent, or opposite each other. Shabbos 6b proves that if the Rabim walk through the walls, a Lechi does not make it a full Reshus ha'Yachid. Tosfos (12b DH Eisivei, 22a) connotes unlike I said, but R. Akiva Eiger said that Tosfos is astounding, due to Shabbos 6b. The Ritva (22a) brings that 'we cannot be Me'arev Reshus ha'Rabim this way' connotes that mid'Oraisa, it does not help. Some Rishonim distinguish based on whether or not it is open to Reshus ha'Rabim, and use this to resolve 94a, which connotes that if one wall fell, one who throws to there from Reshus ha'Rabim is exempt.