The Mishnah begins by outlining a fundamental rule concerning a sukkah: if the covering, known as the skakh, is situated three handbreadths or more away from the sukkah's walls, the sukkah is invalid. This distance refers to the width, meaning the skakh itself does not extend to reach the walls, leaving an open space between the edge of the skakh and the wall. The principle is that any airspace of three handbreadths or more within a sukkah invalidates it, even if this space is at the side rather than in the center.
The Mishnah then presents a second scenario: a house with a hole in its roof, over which skakh has been placed. If the distance from the wall to the sky (i.e., the hole covered by skakh) is four cubits or more, the sukkah is invalid. The reasoning here is that the roof, though made of material suitable for skakh, was originally placed there for the purpose of a house, not for a sukkah. Therefore, it is considered invalid skakh due to the principle of "it was not placed there for its purpose." This means that even if one now intends to use this roof as skakh, it remains invalid because it was not originally constructed with that intention. Consequently, if there is a gap of four cubits or more between this invalid skakh (the house roof) and the actual walls, the sukkah lacks proper walls and is invalid. However, if the gap is less than four cubits, the principle of dofen akumah (a bent wall) is applied. This principle allows us to consider the house walls as extending upwards and bending over to meet the skakh, effectively treating the invalid skakh as part of the wall. In such a case, the sukkah would be valid, though one cannot sit directly under the area treated as a bent wall. Our Mishnah, however, addresses the case of four cubits or more, where dofen akumah cannot be applied, thus rendering the sukkah invalid due to insufficient walls.
A similar principle applies in the Mishnah's third case: a courtyard surrounded by balconies. If skakh is placed over these balconies, and there is a gap of four cubits or more from the skakh to the walls of the courtyard (which serve as the sukkah's walls), the sukkah is invalid. The balconies themselves are considered invalid skakh because they were not erected for the purpose of a sukkah. As in the previous case, if the gap is less than four cubits, the rule of dofen akumah applies, treating the balconies as extensions of the walls, and the sukkah would be valid. Beyond four cubits, however, it is invalid. This case further illustrates the rules of dofen akumah.
The Gemara then asks why the Mishnah needs to present three distinct cases to teach the principle of dofen akumah. It seems redundant. The Gemara explains that each case highlights a unique aspect. In the case of the house with a hole, the walls were originally built for the house. When the house becomes a sukkah, these walls can be considered legitimate sukkah walls, and thus dofen akumah can be applied. However, in the case of the courtyard with balconies, the courtyard walls were built for the homes, not specifically for the balconies. One might think that these walls cannot be extended via dofen akumah to incorporate the balconies as part of the sukkah's walls. Therefore, the Mishnah teaches that even in this scenario, dofen akumah applies, making the sukkah valid if the distance is less than four cubits. The third case involves a large sukkah where invalid skakh, such as metal spits, is used. One might argue that dofen akumah only applies to skakh that is intrinsically valid but becomes invalid due to improper intent (like the house roof or balconies). However, for skakh that is intrinsically invalid, such as metal, one might assume dofen akumah would not apply. The Mishnah teaches that even with intrinsically invalid skakh, dofen akumah can still be invoked.
The Gemara then records a discussion between Rabba and the students of Rav. The students of Rav stated two rules: first, airspace invalidates a sukkah at three handbreadths; second, invalid skakh invalidates a sukkah at four handbreadths. Rabba challenged the second rule, arguing that the Mishnah itself implies that invalid skakh is only invalid at four cubits, not four handbreadths. Rabba understood the Mishnah's ruling regarding the house with a hole (four cubits invalidates) to mean that any invalid skakh of four cubits or more is problematic, regardless of dofen akumah. Conversely, less than four cubits of invalid skakh would not invalidate the sukkah. This interpretation suggests that Rabba does not subscribe to the principle of dofen akumah for sukkah walls. Instead, he views the four-cubit measure as a direct halachic threshold for invalid skakh.
The students of Rav retorted that Rabba misinterprets the Mishnah. They explained that the Mishnah's four-cubit rule for invalid skakh applies specifically to skakh positioned at the side of the sukkah, where dofen akumah can be applied if the distance is less than four cubits. However, in the middle of the sukkah, where dofen akumah is irrelevant, invalid skakh invalidates at four handbreadths. Rabba then presented a hypothetical scenario: if one has less than four handbreadths of invalid skakh next to less than three handbreadths of airspace (both individually valid), and then adds a small amount of intrinsically invalid material to the airspace, making the total invalid skakh (including the added material) four handbreadths, the sukkah would become invalid. Rabba argued that since airspace is more stringent (invalidating at three handbreadths) than invalid skakh (invalidating at four handbreadths), the airspace should combine with the invalid skakh to invalidate the sukkah even before reaching four handbreadths of invalid skakh. He questioned why the more stringent airspace would be treated as less significant than the invalid skakh.
The students of Rav countered that Rabba's question applies equally to his own position. If, according to Rabba, invalid skakh invalidates at four cubits, and airspace at three handbreadths, why would the more stringent airspace not combine with invalid skakh to invalidate the sukkah at a lesser measure? Rabba responded by explaining that for him, the measurements (three handbreadths for airspace, four cubits for invalid skakh) are fixed, divinely ordained measures. Different measurements, even if one is more stringent, cannot combine. Thus, a partial measure of airspace cannot combine with a partial measure of invalid skakh to reach a combined invalidating threshold. However, Rabba argued that according to the students of Rav, who posit that invalid skakh invalidates at four handbreadths in the middle of the sukkah (a measure not explicitly stated in the Mishnah for the middle), this four-handbreadth measure is an arbitrary, significant threshold. If it's merely a significant area, then it should not matter whether that area is composed entirely of invalid skakh or a combination of invalid skakh and airspace.
Abaye then challenged Rabba's assertion that different measurements cannot combine. Abaye pointed out that in a minimum-sized sukkah (seven by seven handbreadths), both three handbreadths of airspace and three handbreadths of invalid skakh invalidate the sukkah. This is because at three handbreadths, the principle of lavud (considering a gap of less than three handbreadths as closed) no longer applies, and the sukkah loses its minimum required dimensions. Since in this context, the invalidating measures for airspace and invalid skakh are equal, Abaye argued that they should be able to combine even in a larger sukkah where their individual invalidating measures differ. Rabba rejected this, stating that in a minimum-sized sukkah, the invalidation at three handbreadths is not due to the equality of measures, but because exceeding three handbreadths of either airspace or invalid skakh causes the sukkah to fall below its minimum dimensions, thus rendering it invalid. It is a matter of insufficient overall sukkah size, not a combination of different intrinsic invalidating measures.
The Gemara then raises another challenge to Rabba's rule against combining different measurements, citing a Mishnah in Massechet Keilim. This Mishnah discusses items susceptible to ritual impurity of midras (impurity imparted by a person with certain ritual impurities sitting, lying, or leaning on an object). Different materials have different minimum sizes for susceptibility to midras impurity: a wool or linen garment at three by three handbreadths, goat's hair at four by four, leather at five by five, and a reed mat at six by six. A braita (a teaching from the Mishnaic period not included in the Mishnah) states that these different items can combine their measures to reach the larger threshold of impurity. For example, a piece of goat's hair garment (requiring four by four) can combine with a piece of wool garment (requiring three by three) to reach four by four and become impure. This demonstrates that different measures can indeed combine. The Gemara resolves this by explaining that all these items share a common minimum measure for impurity of moshav (sitting): one by one handbreadth. Since they share this common measure in one aspect of impurity, they can combine their different measures for other types of midras impurity. This distinguishes it from the sukkah case, where airspace and invalid skakh have no such commonality in their invalidating measures.
The Gemara then revisits the original dispute between Rav and Rabba regarding the measure of invalid skakh in the middle of the sukkah. The first version of the tradition states that Rav held it to be four handbreadths, while Rabba held it to be four cubits. However, a second version of the tradition, from Nehardea, states that Rav held it to be four cubits, whether in the middle or at the side. The Gemara then questions this second version of Rav's opinion, citing a Mishnah that states a plank of four handbreadths width is valid skakh. If Rav held that four cubits of invalid skakh invalidates, then a four-handbreadth plank should certainly be valid. But if Rav held that four handbreadths of invalid skakh invalidates, then this Mishnah would be problematic. The Gemara clarifies that the Mishnah about the plank refers to a plank near the wall, where dofen akumah can be applied, treating the plank as a wall extension, and thus it is valid. However, one cannot sleep under it, as it is considered a wall. This interpretation allows the Mishnah to align with the view that four handbreadths of invalid skakh in the middle invalidates.
The Gemara then brings another challenge from a braita: two sheets of invalid skakh can combine in the middle of a sukkah to invalidate it. However, two planks cannot combine to invalidate the sukkah, according to Rabbi Yehuda. Rabbi Meir, however, holds that planks can combine, just like sheets. The Gemara explains that Rabbi Yehuda does not consider planks to be problematic even at four handbreadths, as he does not make a rabbinic decree against them. Rabbi Meir, however, does make such a decree, fearing that one might confuse a plank with a house roof, and therefore considers planks of four handbreadths or more to be invalid. The Gemara asks, according to the Nehardea version (where Rav holds four cubits invalidates), what case does Rabbi Meir refer to when he says planks combine? If each plank is already four cubits, no combination is needed. If each is less than four cubits, they should be valid skakh. The Gemara answers that Rabbi Meir's statement about combining planks refers to planks on the side of the sukkah. There, planks can combine to reach the four-cubit threshold beyond which dofen akumah cannot be applied, thus invalidating the sukkah by effectively removing a wall. In the middle of the sukkah, however, a single plank of four handbreadths would be invalid on its own, without needing to combine with anything else, according to Rabbi Meir, thus aligning with the Surah version of the dispute.