1) THE SUKAH OF QUEEN HELENA
OPINIONS: The Gemara questions the Amora'im who explain the argument between Rebbi Yehudah and the Rabanan in the Mishnah with regard to a Sukah that is taller than 20 Amos. The Gemara quotes a Beraisa in which Rebbi Yehudah proved his opinion that such a tall Sukah is not Pasul from an incident that involved Hilni the queen. When the elders came to visit Hilni, they found her and her seven sons sitting in a Sukah that was more than 20 Amos tall, and they made no comment about the height of the Sukah. This implies that such a Sukah is valid.
The Rabanan responded that Hilni's Sukah indeed was Pasul, and the reason why the elders did not say anything to her was because a woman is exempt from the Mitzvah of Sukah, and her sons were not yet old enough to be obligated in Mitzvos. Rebbi Yehudah replied that Hilni certainly had a son who was old enough to be obligated in Mitzvos, at least for the sake of Chinuch, and thus Hilni certainly made a valid Sukah for him. It must be that a 20-Amah-tall Sukah is valid.
The Gemara explains that according to Rebbi Yoshiyah (2b, in his comments on the opinion of Rabah), who says that Rebbi Yehudah and the Rabanan argue only when the walls of the tall Sukah do not reach the Sechach, the Beraisa is not problematic because it is the manner of a queen to sit in such a Sukah to enjoy the flow of fresh air. Since this type of Sukah is fit for queens, Rebbi Yehudah and the Rabanan debated whether Hilni's Sukah was valid. However, the Beraisa is problematic according to Rav Huna and Rav Chanan bar Rabah, who explain that the argument between Rebbi Yehudah and the Rabanan applies only to a very small Sukah (such as one that is only seven by seven Tefachim) which is 20 Amos tall. Certainly, a queen would not sit in such a small Sukah. It must be that she sat in a very wide Sukah (which Rav Huna and Rav Chanan bar Rabah say is valid according to both Rebbi Yehudah and the Rabanan), and yet Rebbi Yehudah and the Rabanan still debated whether the Sukah was valid! How do Rav Huna and Rav Chanan bar Rabah understand the Beraisa?
The Gemara answers that although Hilni had a very large Sukah, she sat in a narrow subsection of the Sukah with her sons, and it is that part of the Sukah which Rebbi Yehudah and the Rabanan argue about in the Beraisa.
(a) Rashi points out that this answer does not suffice to justify the opinion of Rav Chanan bar Rabah, who said (2b) that the Rabanan permit a tall Sukah as long as its length and width are large enough to fit a person's head, most of his body, and his table. This size is defined as seven by seven Tefachim. Certainly Hilni's subsection in her Sukah was not that small, because it would not have been possible to fit herself and her seven sons in an area of only seven by seven Tefachim. It must be that Rebbi Yehudah and the Rabanan in the Beraisa argue about a Sukah that is larger than seven by seven Tefachim. This contradicts Rav Chanan bar Rabah's opinion, who says that they agree that such a Sukah is valid. Rashi concludes that the Beraisa indeed is difficult according to Rav Chanan bar Rabah.
(b) TOSFOS answers that even according to Rav Chanan bar Rabah, Hilni's Sukah would have been invalid according to the Rabanan. Even though the Sukah certainly was not a small Sukah of seven by seven Tefachim (which would not fit Hilni and her sons), perhaps Hilni and her sons sat in a long and narrow subsection of the Sukah. If that compartment was less than seven Tefachim wide, it could have been any length and still be invalid according to the Rabanan.
It appears that Rashi and Tosfos argue about the definition of the minimum size of a Sukah. Rashi understands that the measurement of "seven Tefachim by seven Tefachim" (16b, and Rashi to 4b and 16a) describes the minimum area of a valid Sukah. As long as the Sukah's area is equal to the area of a seven-by-seven Sukah (49 square Tefachim), then it is valid, such as a Sukah three Tefachim long by 16 1/3 Tefachim wide. Accordingly, a Sukah that has the minimum measurements cannot contain a mother and her seven children.
Tosfos, on the other hand, maintains that when the Gemara says that a Sukah must be seven Tefachim long, it means that it must have a minimum of seven Tefachim in each dimension, in length and in width. It does not matter whether the area of the Sukah comes to 49 square Tefachim; if any one dimension does not reach seven Tefachim, it is invalid. Accordingly, a Sukah that does not have the minimum measurements can contain a mother and her seven children, such as a Sukah that is very long but is less than seven Tefachim wide.
(In most other Halachic matters that involve a minimum area, such as the four-by-four-Amah minimum requirement for a Sukah according to Rebbi, or the measure of Mechitzos of four Tefachim high by four Tefachim long, the assertion of Tosfos certainly applies. Both dimensions must be a full four Amos or Tefachim. Perhaps Rashi differentiates between the minimum size of a Sukah and the measurements of other Halachic matters, because the minimum size of a Sukah depends on a practical consideration: one must be able to fit a table along with a person into a Sukah. The table may be placed either at the person's side, or in front of him, and thus as long as the area can contain a person and a table (49 square Tefachim), the Sukah is valid.)
2) HALACHAH: THE MINIMUM SIZE OF A SUKAH
OPINIONS: The Gemara records two arguments between Beis Shamai and Beis Hillel. The first argument involves the minimum size of a Sukah. Beis Shamai says that a Sukah must be at least seven by seven Tefachim (large enough to hold a person's head, most of his body, and his table). Beis Hillel says that the minimum size is only six by six Tefachim (large enough to hold a person's head and most of his body, without his table).
The second argument involves a person who sits in a Sukah while his table is outside of the Sukah. According to Beis Shamai, the Rabanan decreed that he does not fulfill his obligation in such a case, because of the concern that he might be drawn after his table and eat outside of the Sukah. Beis Hillel maintains that the Rabanan made no such decree.
In both cases, Beis Shamai's opinion is the more stringent one. According to Beis Shamai, is the Sukah in each case invalid mid'Oraisa or mid'Rabanan? In addition, what is the Halachah in each case?
(a) The RIF (13a of the pages of the Rif) writes that in both cases, the Halachah follows the view of Beis Shamai, because the rulings in the two cases "depend on the same reason." The RAN and the RITVA explain that the Rif maintains that the Pesul of a small Sukah is mid'Rabanan, just like the Pesul of a large Sukah with the table outside. In both cases, the Rabanan decreed that the Sukah is invalid, lest the person be drawn after his table. Since the Rif says that both types of Sukah (a small one, and one with the table outside) are invalid for the same reason (because of the enactment of the Rabanan), he clearly maintains that Beis Shamai invalidates them only mid'Rabanan.
The MISHNAH BERURAH (OC 634, in Bi'ur Halachah DH Afilu) asks that although the Rif implies that a small Sukah is invalid only mid'Rabanan, whenever the Gemara discusses the minimum size of a Sukah (see, for example, 7a and 16a-b) it seems to take for granted that there is no minimum size mid'Oraisa other than seven by seven Tefachim. If that requirement is mid'Rabanan, then the Gemara should make some mention of the size required mid'Oraisa.
To answer this question on the Rif, the Mishnah Berurah (in Sha'ar ha'Tziyun #7) suggests that perhaps the Rif means to say that a small Sukah is invalid mid'Oraisa. How does the Mishnah Berurah explain the Rif's statement that both a large Sukah with the table outside and a small Sukah are invalid for the same reason?
The Rif apparently understands that the fact that the Torah invalidates a small Sukah (the size of which is not large enough to contain one's table) reveals that the Torah itself is concerned that if the table is outside of the Sukah because of its small size, one might be drawn after the table. From that Torah law, the Rabanan deduced that they should enact a Gezeirah to invalidate a large Sukah when the table is outside, even though the Torah does not invalidate such a Sukah (since it is possible for the table to be inside the Sukah; see RAN).
(b) However, the ROSH, RAN, and others explain that the two arguments are completely unrelated. Rav Shmuel bar Yitzchak, who rules like Beis Shamai, says that a Sukah must be large enough to contain "his head, most of his body, and his table." He rules like Beis Shamai only with regard to the minimum size of a Sukah.
Beis Shamai's reason in the case of a large Sukah with the table outside is different. In that case, Beis Shamai maintains that the Rabanan invalidated the Sukah out of concern that a person might be drawn after his table. Rav Shmuel bar Yitzchak does not say that the Halachah follows Beis Shamai in that case. Consequently, a Sukah less than seven by seven Tefachim is invalid mid'Oraisa, while a large Sukah with the table outside is valid mid'Oraisa. Even according to Beis Shamai, a large Sukah with the table outside is invalid only mid'Rabanan.
The MISHNAH BERURAH (OC 634, end of Bi'ur Halachah DH Pesulah) suggests that this is also the opinion of the RAMBAM (Hilchos Sukah 4:1, 6:8).
(c) TOSFOS (3a, DH d'Amar Lach) and TOSFOS RABEINU YEHUDAH HE'CHASID (Berachos 11a) suggest that in both cases, the Sukah is invalid mid'Oraisa according to Beis Shamai. When the Gemara says that a large Sukah with the table outside is invalid because of the decree of the Rabanan, it means that the Rabanan decreed that when a person sits in his Sukah with the table outside he does not fulfill the Mitzvah of Sukah even mid'Oraisa. The Rabanan have the authority to decree that a person who does not perform a Mitzvah in the way that they prescribed does not fulfill even the Torah obligation.
How, though, are the Rabanan able to prevent one from fulfilling his obligation mid'Oraisa? Perhaps a person performs every Mitzvah "according to the will of the Rabanan" (as the Gemara says with regard to the act of Kidushin, "Kol d'Mekadesh a'Da'ata d'Rabanan Mekadesh," Kesuvos 3a). The will of the Rabanan is that the act not be considered the fulfillment of a Mitzvah when it is done in a way contrary to the way the Rabanan prescribed. Even if one fulfills a Mitzvah when he does not have specific intent (Kavanah) to fulfill it, a person who specifically intends not to fulfill a Mitzvah does not discharge his obligation (as TOSFOS rules in Menachos 40b, DH mi'Mai).
HALACHAH: The Halachah follows the view of Beis Shamai in the case of a small Sukah, because Rav Shmuel bar Yitzchak rules like Beis Shamai in that case. In the case of a large Sukah with the table outside, TOSFOS, the ROSH, and many other Rishonim rule like Beis Hillel. Since Rav Shmuel bar Yitzchak does not discuss that case (according to their understanding of his words), the Halachah follows Beis Hillel, because of the general principle that whenever Beis Shamai and Beis Hillel argue the Halachah follows Beis Hillel.
However, the RIF and RAMBAM rule like Beis Shamai even in the case of a large Sukah with the table outside, because they maintain that both cases depend on the same reasoning (see (a) above). Hence, if the Halachah follows Beis Shamai in one case, then it must follow him in the other case as well. This is how the SHULCHAN ARUCH rules (OC 634:4).