What are Beis Shammi and Beis Hillel really arguing about in Succah Daf 3a when the Mishna states: מי שהיה ראשו ורובו בסוכה ושולחנו בתוך הבית ב"ש פוסלין ובית הלל מכשירין?
They are not arguing regarding the 1) smallest size succah (6x6 and 7x7 tefachim) or 2) the manner in which a person eats in a succah. They are arguing about one point only in both a small and large succah - namely what gives a succah the attribute of “Dwellability”.
Daf 7b list 8 Tannaim that held that while a succah is a temporary dwelling, a succah must contain some aspect of permanence similar to a house in order to have “Dwellability”. While all agreed that a succah must have a fixed attribute none of the Tannaim agreed as to the particular aspect. Rebbi held that a succah needs to be 4x4 amos to be valid. While Beis Shammi held that a succah needs to have a table inside in order to be valid. Beis Hillel is not listed among the Tannaim that held that a succah needed an aspect of permanence to be a valid succah. Still from Daf 3a we find that Beis Hillel does holds that a succah needs “Dwellability”, albeit not from an aspect of permanence. And according to Beis Hillel this is achieved with just ראשו ורובו, while Beis Shammi holds ראשו ורובו ושולחנו.
What are their rationales? All hold that the Torah requires us to dwell in a succah. Therefor the succah must be a dwelling and not like other structures such as a hut or guard tower where one does not dwell. Rebbi held that a succah is a dwelling like a house which if built less than 4x4 amos will lose its Dwellability. As the Baraisa states on Daf 3a regarding the laws of houses, רבנן מודו דאי אית ביה ד' אמות על ד' אמות דיירי ביה אינשי ואי לא לא דיירי ביה אינשי. Beis Shammi held that the unique feature of a dwelling is that you eat meals at a table, otherwise it is not a dwelling but more like a hut or other type of structure but not a dwelling. Beis Hillel, however held that even if a person simply sat in the succah and ate from their hand it was enough to make it a dwelling. Therefore, Dwellability is the argument between Beis Shammi, Beis Hillel and Rebbi.
One might say wait a minute we learned in the Gemara that Rav Nachman bar Yitzchak proposed that they were arguing over the manner in which they ate. According to this case Beis Shammi prohibits lest he be draw outside of his succah by the table and not fulfill the mitzvah of succah and Beis Hillel does not prohibit. However, this is not supported by the Gemara’s explanation of the Mishna when it states: מי שהיה ראשו ורובו בסוכה ושולחנו בתוך הבית ב"ש אומרים לא יצא וב"ה אומרים יצא. The Gemara uses the terms לא יצא and יצא instead of Rav Nachman bar Yitzchak’s language of גזרינן and לא גזרינן associated with the proposed case regarding the manner in which one eats. This switch should be enough of an indication that it can't be going on the manner in which he eats, but if not then from fact that it forces the one holding this position to conclude that a Rabbinic decree to prevent someone from eating outside his succah will result in nullifying a Biblical mitzvah. This is unacceptable when they could have just kept the language of גזרינן and לא גזרינן.
Still if that is not enough, the singular focus of the Mishna on the concept of Dwellability can be concluded directly from the Mishna itself that states: מי שהיה ראשו ורובו בסוכה ושולחנו בתוך הבית ב"ש פוסלין ובית הלל מכשירין. Why does the Mishna state: בתוך הבית? We are talking about a succah so just state that the table is outside the succah. Why does the table have to be inside the house? This is the Mishna’s inclusion of Rebbi’s opinion with Beis Shammi and Beis Hillel regarding the requirement for Dwellability. And since he holds that dwellability is obtained at 4x4 amos it can be achieved in one of two ways 1) either by a home owner removing the roof of a corner of his house the size of ראשו ורובו ושולחנו or ראשו ורובו and placing s’chach there or 2) having a stand-alone succah that by itself has 4x4 amos to it. And this is why the first Baraisa on Daf 3 states: מחזקת ראשו ורובו ושולחנו כשרה רבי אומר עד שיהא בה ארבע אמות על ארבע אמות. The identification of “בה” is left undefined because it can be either a small succah in a house of 4x4 amos or a stand-alone succah of 4x4 amos. In the first case, Rebbi holds that just like a small room which is less than 4x4 amos gets its dwellability from the total house structure, so too would a small succah inside the house obtain its dwellability from the total house as well and therefore can be smaller than 4x4 amos. And this is why the second Baraisa that does specifically speak about the succah itself states: רבי אומר כל סוכה שאין בה ארבע אמות על ארבע אמות פסולה וחכ"א אפילו אינה מחזקת אלא ראשו ורובו כשרה. On a parallel level the Gemara brought both of these Baraisa’s in addition to the Mishna in order for us to know that Rebbi holds that you must have 4x4 amos for dwellability because you might have thought that perhaps if the succah had a table or you simply sat in it with ראשו ורובו, Rebbi would say that too could provide dwellability. Comes both of these Baraisa’s to teach that Rebbi held that the only way to achieve dwellability is with 4x4 amos.
Now, when the Mishna states: מי שהיה ראשו ורובו בסוכה ושולחנו בתוך הבית ב"ש פוסלין ובית הלל מכשירין most people continue the line of thought of the proposed solution by Rav Nachman bar Yitzchak of a case where a succah is adjacent a house and he is sitting at the edge of his succah and the table is inside the house. Beis Shammi says invalid and Beis Hillel says valid. From this case, we could deduce that Beis Hillel holds that a succah doesn’t obtain dwellability from Beis Shammi’s table or Rebbi’s 4x4 amos, but from ראשו ורובו alone. We would also deduce that Beis Shammi does not obtain dwellability from Beis Hillel’s ראשו ורובו alone, but we would not be able to conclude that they hold that dwellability can only be obtain exclusively through the table apart from or separately with Rebbi’s 4x4 amos. Therefore, the Mishna adds the words מי שהיה to create and indicate that there are two cases from which we should fully deduce both Beis Shammi and Beis Hillel’s opinions.
It works like this. If the Mishna wanted us to learn out both positions from one case ראשו ורובו בסוכה ושולחנו בתוך הבית ב"ש פוסלין ובית הלל מכשירין it would not have been enough. So they added מי שהיה for us to read and think of it like this: “In a case where מי שהיה ראשו ורובו בסוכה ושולחנו בתוך הבית ב"ש פוסלין ובית הלל מכשירין, but what if it was a different case? And what case could it be?” The addition מי שהיה forces us to conclude that the only other possible case that can be made from the Mishna is Rebbi’s second case where a small succah is made inside a house and the table is necessarily inside the house but outside the succah. This answers the question about why the Mishna states that the table is בתוך הבית instead of simply stating outside the succah. And now that we realized that this is the second case driven by מי שהיה we can conclude that in a case where the succah is made from a room inside the house by removing part of the roof and the table is outside the succah, Beis Shammi holds the succah is invalid which enables us to deduce that the 4x4 amos of the house does not provide dwellability in the absence of the table and that the table is exclusively the answer to provided dwellability according the Beis Shammi. This is the case because there are no other options. You can either take ראשו ורובו out of the succah (which makes no sense) or put the table inside the succah. But now regarding Hillel, the opposite is the case you can’t fully deduce their position. Beis Hillel rules valid but you can’t conclude if they hold ראשו ורובו exclusively or it’s valid because it is inside the Rebbi’s house of 4x4 amos without which it would be invalid. Hence, both cases are needed and this is the reason why מי שהיה was added. Not because of a concern about the manner in which a person eats.
In conclusion, this demonstrates that when the Gemara stated וחסורי מיחסרא והכי קתני, at least in this instance, it did not mean the Mishna is lacking but that our understanding of the Mishna is lacking and therefore the Gemara has to expand it into two parts. And now that they did this we can once again use the Mishna as a much more effective and concise tool for memorization of all the opinions and their numerous details.
Furthermore, it is proof that the Mishna and Baraisas are not at all speaking directly about a minimum size of a succah. Rather, only about the requirements for “Dwellability” of a succah which tangentially reveals the minimum size of a succah as ראשו ורובו ושולחנו and ראשו ורובו. Therefore the minimum size succah is a person-specific size generally discussed as 7x7 or 6x6 tefachim. However, it could be even less, as deduced from the incident of Beis Shammi the Elder's baby grandson in Daf 28a brought by the Mishna to further define the parameters of the argument after the incident R’ Yochanan ben HaChoranis dwelling without a table. To which Beis Shammi is assumed to have held that when a baby normally dwells in a house it does not eat at a table therefore in this specific case they would agree with Beis Hillel that ראשו ורובו is correct.