A MAJORITY OF KOSHER SECHACH IS REQUIRED [Sechach:majority]
Mishnah: If there is more shade than sun it is Kosher.
22b - Inference: If the shade and sun are equal it is Pasul.
Contradiction (Mishnah): If there is more sun than shade it is Pasul.
This implies that if the shade and sun are equal it is Kosher!
Answer: One Mishnah discusses above (by the Sechach), and one discusses below (on the ground).
Rav Papa: This is like people say, a Zuz above is like Istira below (these are coins of known sizes).
19a - Beraisa: Pesel that leaves a Sukah is considered a Sukah.
R. Yochanan: This teaches about a Sukah in which the majority is mostly shade and the minority is mostly sun. One might have thought that where it is mostly sun it is Pasul. The Beraisa teaches that this is not so;
The Beraisa says that it 'leaves' the specifications for Hechsher Sukah (it does not have mostly shade).
Rif and Rosh (2:2): If the shade and sun are equal below it is Kosher, for we know that there is more shade above. If the shade and sun are equal above it is Pasul, for we know that there is more sun below.
Question (R. Tam, brought in Rosh, ibid. and Ba'al ha'Ma'or): Rav Papa himself permits Parutz k'Omed (he should permit when there is half shade above)! Also, Istira is half a Zuz (this shows that there is less sun below than above)!
Answer (R. Tam, ibid.): Rather, Rav Papa teaches that if the shade and sun appear equal to one standing above (near the Sechach) it is Kosher, for Parutz k'Omed is permitted. If they appear equal to one standing below (on the ground) it is Pasul, because a gap above like a Zuz appears (smaller) like an Istira below. We know that stars are big in Shomayim, but they appear small to us.
Rebuttal #1 (Hasagos ha'Ra'avad): It is unreasonable that an extra Mishnah was taught to teach about one who measures inaccurately from far away!
Rebuttal #2 (Milchamos Hash-m): A tradition from Sinai permits Parutz k'Omed regarding Pasul Sechach and walls for Shabbos. It cannot apply to gaps in the Sechach, for then there will be more sun below, and the Sukah is not "L'Tzel Yomam me'Chorev". Rather, at least half shade below is required. The Rif does not explicitly say that Istira is bigger than a Zuz. Perhaps Rav Papa refers to the size of the shade! Istira can refer to a Sela (four Zuz) or half a Zuz. Normally, it refers to four Zuz. Sechach looks smaller from below but the sun does not, just like a man on a mountain looks tiny from afar, but a fire appears bigger. If the Sukah is tall, one may err more. If from below it appears like there is slightly more Sechach, perhaps there is really a minority!
Rebuttal #3 (Rosh, ibid.): Parutz k'Omed applies to Pasul and Kosher Sechach. However, regarding air a tradition from Sinai requires more shade. We find that vertical sticks within three Tefachim of each other helps for walls, even though Parutz Merubah Al ha'Omed, but it does not help for Sechach because there is more sun. Even though an Istira is worth half a Zuz, perhaps it is of copper and is bigger than a (silver) Zuz.
Ran (DH Gemara): There are Sela'im and Zuzim of Tzuri and of 'Medinah' (standard coins). In both cases Istira is a Sela, i.e. four Zuz; Tzuri coins are worth eight times as much. (Rambam Hilchos Ishos 10:8 - they are pure silver. Medinah coins are one eighth silver and seven eighths copper.)
R. Mano'ach (Hilchos Sukah 5:19): Even if an Istira is half the weight of a Zuz, perhaps it is wider but thinner. We are concerned only for sun that comes through the roof, not through the walls, because it is normal to have windows in the walls, but not in the roof.
Rosh (ibid.): Ba'al ha'Itur says that if the gap between reeds of Sechach equals the width of the reeds and there is more shade below, it is Kosher. If the shade and sun are equal below and there is more sun above, it is Pasul.
Rambam (Hilchos Sukah 5:19): If there are gaps in the Sechach through which air can be seen:
If there is as much air as Sechach it is Pasul, for there is more sun than shade. If there is more Sechach than air it is Kosher.
Rambam (20): If the majority of the Sechach was mostly shade and the minority was mostly sun, since overall it is mostly shade it is Kosher.
Rosh (1:35): The Halachah is like all of the Amora'im who explained the Beraisa of Pesel that leaves a Sukah. R. Yochanan teaches that if the majority has two Mashehu's (tiny amounts) more shade than sun, and the minority had Mashehu sun more than shade, so overall there is Mashehu more shade, it is Kosher.
Ran (9a DH Garsinan): Perhaps this refers to a minimal Sukah of seven by seven. But if the Sukah is bigger and there is an area seven by seven in which there is more sun, it is not clear whether or not it is Kosher due to the rest of the Sukah.
Shulchan Aruch (OC 631:1): If a Sukah has equal amounts of sun and shade above it is Pasul, for the sun spreads over distance, and it is more sun below. If it has equal amounts of sun and shade below it is Kosher.
Bach (DH Ha): The Tur did not bring R. Tam's opinion because the Rosh refuted it. He did bring the Ba'al ha'Itur's opinion. We should be stringent for both of these, and also for R. Tam's opinion. Therefore, if there is an equal amount of shade on top or on bottom or (on top) when viewed from the ground, it is (Safek) Pasul.
Korban Nesan'el (9): When the gaps between reeds equals the width of the reeds, will the Ba'al ha'Itur argue with the others about the amount of shade on bottom?! This can be easily verified! Rather, when the sun is overhead there is more sun below. When the sun is on an angle there is more shade below. The Sukah must have enough Sechach to be Kosher at all times.
Bi'ur Halachah (DH she'Ein): The Shulchan Aruch did not bring the Ba'al ha'Itur's opinion because the Rif, Rambam and Rosh all disagree.
Kaf ha'Chayim (2): Rashi agrees with R. Tam that if it there is about as much air as Sechach, one may not look from below to gauge which is the majority. However, if one sees from below that there is much more Sechach, this suffices.
Igros Moshe (OC 1:182): Rashi and the Rambam disqualify when there are equal amounts of sun and shade above. R. Tam asks, this is Parutz k'Omed, which is permitted! The Ran and Rosh answer that we are concerned for shade and sun on bottom. The Kesef Mishneh (Sukah 5:16) holds that the Rambam agrees with this.
Igros Moshe (ibid.): The Gemara connotes that a Sukah will never have more shade above and more sun below. (In such a case the contradiction would remain!) Surely, this is the truth. If we observe otherwise, there must be another reason, e.g. the Sechach consists of cylindrical reeds which allow more sun through (because the sun is not directly overhead), just like when different parts of the Sechach are at different heights. If it is known that there is more Sechach above one need not be concerned.
Shulchan Aruch (2): If in the majority of the Sukah there were two Mashehu's more shade than sun, and the minority had Mashehu sun more than shade, so overall there is Mashehu more shade, it is Kosher.
Darchei Moshe (1): R. Yerucham says that if in the minority of the Sukah there is more shade, and in the majority there is more sun, even if overall there is more shade, it is Pasul. It seems that the Rosh and Tur agree with this.
Rema: Some are stringent if there is a place seven by seven in which there is more sun, even if overall there is more shade.
Mishnah Berurah (4): The first opinion says that the part with more sun is Batel. The latter opinion it stringent to say that it is not Batel. All agree that where there is more shade it is Kosher, since it has Shi'ur Hechsher Sukah. However, if there are only two walls around the area with more sun perhaps the entire Sukah is Pasul.