GEMARA

17a - Mishnah: If the Sechach was three Tefachim (horizontally) from the walls, it is Pasul. If the roof caved in and one put Sechach in its place, if there are four Amos from the wall to the Sechach it is Pasul.

Gemara - Version #1 (Sura) Rabanan of Bei Rav: Three Tefachim of air (a gap in the Sechach) of disqualifies a Sukah. Four Tefachim of Pasul Sechach disqualifies.

Rabah: Surely, you learn that the Shi'ur for air is three Tefachim from the Reisha. You should learn from the Seifa that the Shi'ur for Pasul Sechach is four Amos!

Rabanan: Rav and Shmuel explained that the Seifa is due to Dofen Akumah (we consider the wall to be bent and to meet the Kosher Sechach).

17b - Version #2 (Neharda'a) Shmuel: Four Tefachim of Pasul Sechach disqualifies in the middle. Four Amos of Pasul Sechach disqualifies from the side.

Rav: In either case the Shi'ur is four Amos.

18a - Abaye: If there was three Tefachim of air in a big Sukah and one diminished it, through reeds or spits (Kosher or Pasul Sechach), this is a good Mi'ut (the Sukah is Kosher);

In a small Sukah one may diminish with reeds, but not with spits.

This refers to from the side. Rav Acha and Ravina argue about whether or not Lavud applies in the middle.

19a - Beraisa: Pesel that leaves a Sukah is considered a Sukah.

R. Oshaya: This teaches about less than three Tefachim of Pasul Sechach in a Sukah. The Beraisa says that it 'leaves' the guidelines for (Sechach for) a Sukah. (Other Amora'im explain the Beraisa differently.)

Question (R. Hoshaya): What is the Chidush? It is no worse than air, and less than three Tefachim of air does not disqualify (even) a small Sukah!

Answer (R. Aba): Pasul Sechach joins (to complete the Shi'ur for a Sukah) and one may sleep under it. Air joins, but one may not sleep under it.

RISHONIM

Rif: The Halachah follows Shmuel, who says that the Shi'ur for Pasul Sechach is four Tefachim in the middle and four Amos from the side. This is because this is like Rav in the first version. This refers to a big Sukah. In a small Sukah the Shi'ur is three Tefachim in the middle or on the side.

Ran (before DH ul'Inyan): In a small Sukah of seven Tefachim there may not be three Tefachim of Pasul Sechach, for then we cannot apply Lavud.

Rif and Rosh (1:33): One may diminish three Tefachim of air in a big Sukah through reeds or spits. In a small Sukah one may diminish with reeds, but not with spits.

Rif: This refers to (three Tefachim of air) from the side. Rav Acha and Ravina argue about whether or not Lavud applies in the middle (to allow up to three Tefachim). The Halachah is that we say Lavud, for wherever Rav Acha and Ravina argue Ravina is more lenient and the Halachah follows him.

Rambam (Hilchos Sukah 5:13,14): If one used Pasul and Kosher Sechach, if there was three Tefachim of Pasul Sechach in one place, whether in the middle or on the side, it is Pasul. This refers to a small Sukah.

R. Mano'ach: Some say that four Tefachim of Pasul Sechach disqualify a small Sukah. We do not rely on them.

Rambam (14): In a big Sukah four Tefachim of Pasul Sechach disqualify in the middle.

Rambam (Hilchos Sukah 5:20): If there were three Tefachim of air in one place, whether in the middle or on the side, it is Pasul.

Rambam (ibid.): One must diminish the air to be less than three Tefachim. If one diminished it through pillows and blankets, if it is a big Sukah it is Kosher. If it is a small Sukah it is Pasul, unless one diminished with Kosher Sechach.

Rosh (ibid.): Air and Pasul Sechach disqualify through three or four Tefachim when they run from the middle wall along the entire Sukah to the opening, for then it separates the Kosher Sechach into two parts, each of which has only two walls. If the Pasul Sechach goes (parallel to the opening) from one side wall to the other, if a Shi'ur Sukah remains between it and the middle wall, the inner part of the Sukah is Kosher. If all the Pasul Sechach is within four Amos of the middle wall, even the outer part of the Sukah is Kosher through Dofen Akumah. If there is merely three by three Tefachim of air or four by four Tefachim of Pasul Sechach it does not disqualify in any direction.

Korban Nesan'el (200): However, one may not sleep under that place.

Ran (DH ul'Inyan): The Ba'al ha'Ma'or allows one to sleep under Pasul Sechach if it is less than four (by four) Tefachim. The Ra'avad permits only if it is less than three (by three) Tefachim. Less than three Tefachim of air does not disqualify, but one may not sleep under it

Rosh (ibid.): Pasul Sechach and air do not join, for their Shi'urim are different. If there are two areas of two Tefachim each of Pasul Sechach and less than three Tefachim of air in between, it is not clear whether or not the Pasul Sechach on the two sides join.

Korban Nesan'el (400): Even though we say Lavud even when it is a stringency, we do not consider that the area in between is Pasul Sechach. Alternatively, we say Lavud to be stringent on the side. In the middle the Halachah follows Ravina to be lenient, so we do not say Lavud to be stringent.

Rosh (1:32): The Rif rules like Shmuel because this is like Rabanan of Bei Rav in the first version. BaHaG rules like Rav in the second version. Ba'al ha'Itur says that the Yerushalmi supports BaHaG, for in it R. Chiya teaches that the Shi'ur is four Amos. Presumably, BaHaG is correct. We cannot support Shmuel from Rabanan of Bei Rav, for the second version says that the first version is mistaken (and Rav actually argued with Shmuel)!

It seems that just like regarding air there is no distinction whether it is in the middle or from the side, also regarding Pasul Sechach. However, Meforshim argue about this, so we are stringent. The Ramban says that in many places the Stam Gemara holds that we say Dofen Akumah. This shows that the Shi'ur for Pasul Sechach in the middle is four Tefachim, for according to the opinion that it is four Amos even in the middle, this is the Shi'ur, and even from the side it is not because of Dofen Akumah. Also the Rambam and R. Yeshayah rule like this.

Rosh (1:35): The Halachah is like all of the Amora'im who explained the Beraisa of Pesel that leaves a Sukah. R. Oshaya teaches that Pasul Sechach joins to complete the Shi'ur for a Sukah and one may sleep under it.

Question: Every Sukah has holes (in the Sechach). How can one be careful not to sleep under air?

Answer: One may not sleep under air only if it runs along the entire length of the Sukah.

Ran (9a DH Garsinan): Some say that since Lavud applies to air less than three Tefachim, it is as if the entire area has Sechach, so one may sleep under it, but one may not sleep under Pasul Sechach. This is wrong. The Gemara assumed that air is more stringent than Pasul Sechach. It does not seem like it retracted. Rather, one may not sleep under air. This is only if it is Rosho or Rubo can fit underneath. If it is less than this it is permitted. One need not make the Sechach as solid as the roof of a house.

POSKIM

Shulchan Aruch (OC 632:1): Four Tefachim of Pasul Sechach disqualifies in the middle.

Beis Yosef (DH Sechach): Rashi says that four Tefachim is a significant Shi'ur. It is as if there are two Sukos (one on each side).

Shulchan Aruch (ibid.): Less than this is Kosher, and one may sleep under it.

Source (Gra DH u'Mutar): R. Meir disqualifies planks more than three Tefachim wide, but he agrees that if there was space just as wide between them with Kosher Sechach, it is Kosher (18a).

Mishnah Berurah (3): One should be stringent like the opinion that forbids sleeping under it if it is three Tefachim or more.

Taz (3, citing Levush): The Pasul Sechach is Batel to the Kosher Sechach and becomes Kosher, like any Isur that is Batel in Heter. This is difficult, for Bitul does not apply to something recognizable. Perhaps instead it is a tradition from Sinai. Alternatively, perhaps mid'Oraisa Bitul applies even when the Mi'ut is recognizable. We learn from following the majority opinion of the Sanhedrin; there, the minority is known. Normally, mid'Rabanan one must discard the Isur if it is known. Regarding the Mitzvah of Sukah Chachamim did not require this.

Rebuttal (Taz 3): Tosfos (Chulin 95a DH Sefeiko) says that mid'Oraisa Bitul is only when the Isur is mixed and not recognized. The minority opinion on the Sanhedrin is ignored, it is not transformed to the opposing opinion. Pasul Sechach does not become Kosher. However, if it is small, we ignore it.

Shulchan Aruch (ibid.): This refers to a big Sukah with seven by seven Tefachim without the Pasul Sechach. In a small Sukah that is only seven by seven Tefachim, it is Kosher only if the Pasul Sechach is less than three Tefachim. One may sleep under it.

Magen Avraham (3): This is not precise. Just like a big Sukah needs seven Tefachim of Kosher Sechach without the Pasul Sechach, also a small Sukah.

Rebuttal (Chazon Ish OC 144:3): Since less than three Tefachim is not significant, perhaps this amount of Pasul Sechach can join for the Shi'ur Sukah.

Air disqualifies a big or small Sukah only if there are three Tefachim, whether this is in the middle or from the side. One may not sleep under it.

Taz (4): We are more stringent about air than Pasul Sechach because the break in the Sechach is more evident.

Rema: This is only if it (less than three Tefachim of air) runs along the entire Sukah, or if it is enough for Rosho v'Rubo to fit under it.

Mishnah Berurah (12): The Ran and Ritva forbid to sleep under it if it fits Rosho or Rubo.

Rema (ibid.): Otherwise it is Kosher, for every Sukah has holes (in the Sechach). Air and Pasul Sechach disqualify through three or four Tefachim when they split the Sukah and do not leave a Shi'ur Hechsher Sukah with walls in one place. If a place with Shi'ur Hechsher Sukah remains in one place it is Kosher, and also other places that are connected to it through the sides.

Mishnah Berurah (16): The Magen Avraham says that other places that are connected can join to the Shi'ur Sukah. The Bechori Yakov is hesitant about this.