Before us is the eleventh Mishnah in the first chapter of Maseches Eduyos, the last of the Mishnayos in which Shammai and Beis Shammai disagree. Its subject is the laws of tumah and taharah, and it includes two separate scenarios: a bride's chair, and a seat that was fixed into a kneading trough.
Background: What is a vessel that is susceptible to tumah?
For something to be susceptible to tumah, it must be a vessel that serves a particular purpose. That purpose is generally determined by the intention of the person who made the vessel. Another point must be introduced, one that applies throughout Seder Taharos: when the Mishnah says about an object that it is "tamei," it does not mean that the object is actually impure, but rather that it is susceptible to tumah.
The first scenario: a bride's chair:
A bride's chair is an ornamented chair, fit only for a bride to sit in; a woman who is not a bride would be embarrassed to sit on it. (It is worth looking at a picture of such a chair, since a picture is worth a thousand words.) The Mishnah deals with a case in which the chair's coverings were removed - the armrests on its sides broke off. Now the chair is not suitable for a woman who is not a bride, since she would not sit in a bride's chair, and it is not suitable for the bride herself either, since it is not dignified to sit in a broken chair. The question, then, is whether such a chair is still susceptible to tumah, even though it can no longer serve its original purpose.
The words of the Mishnah: "Kisei shel kallah shenitlu chipuyav, Beis Shammai metam'in u'Beis Hillel metaharin" - a bride's chair whose coverings were removed, Beis Shammai rule that it is susceptible to tumah and Beis Hillel rule that it is not. This Mishnah appears as well in the twenty-second chapter of Maseches Keilim.
Beis Shammai rule that it is susceptible: The chair is still susceptible to tumah. True, it is no longer suitable for a bride, but any ordinary person can sit on it, and it functions as a chair in every respect. The loss of the armrests does not nullify its use.
Beis Hillel rule that it is not susceptible: The chair was made from the outset for a bride. Once it broke, the bride no longer sits on it, and it is therefore a chair that does not function and requires repair. No one sits on it until it is fixed, and therefore it is not susceptible to tumah.
Shammai, the strictest of them all, says: "Af malben shel kisei tamei": chairs are made for sitting, and as long as the object can be used for sitting it is susceptible to tumah - even if nothing remains of it but the basic frame. That frame, something like a stool or even less - a square with four legs coming out from underneath it - is fit to sit on, and therefore it alone is already susceptible to tumah.
The halachah follows Beis Hillel: a chair that broke and is not fit for its ordinary use, and which no one uses, is not susceptible to tumah at all.
Two types of susceptibility to tumah:
The second half of the Mishnah is built on this foundation, and to understand it we must first know that in the laws of tumah there are two categories of vessels. The first category is general vessels, which become tamei through direct contact, or by being under one roof with a corpse (ohel), and the like. The second category is a special subcategory: mishkav, moshav and merkav - items designed to bear a person's weight, such as a chair, a bed, a rug or a saddle. These vessels receive a special tumah from four powerful sources of tumah: the zav, the zavah, the niddah and the yoledes, who are tamei on account of fluids that issue from them. These four transmit tumah in a manner called midras, that is, by placing their weight on something designed to bear a person's weight.
Tumas midras has two novel features beyond the ordinary rules of how tumah spreads:
Direct contact is not required: like the story of the princess and the pea - if ten mattresses are lying one on top of the other, and a yoledes, niddah or zavah sits on top of them, all the mattresses become tamei, including the bottom one, even though she never touched it at all.
The vessel becomes an av hatumah: ordinarily, when tumah is transmitted, each vessel in the chain drops one level below the one before it, but with midras this is not the case. If a zav, zavah, niddah or yoledes sits on a chair, and the chair is standing on a rug, both the chair and the rug become avos hatumah, and they transmit tumah to other vessels and to people.
It follows that with every utensil one must clarify: is it defined as something meant to bear a person's weight, and therefore susceptible to midras tumah, or is it an ordinary utensil, which is not susceptible to midras and becomes tamei only through direct contact? There is a further principle to introduce: when two parts are joined into a single utensil, it is the primary, functional part that determines its status. Thus a knife whose blade is metal and whose handle is wood is considered entirely a metal utensil, since the wood is secondary to the blade, which is the essence of the knife.
The Second Case: A Chair Fixed Into a Kneading Trough
The Mishnah discusses an unusual case: a stool, which is a utensil susceptible to midras tumah because it is meant to bear a person's weight, that was attached to a kneading trough. An areivah is the basin in which a woman would knead her dough in the times of the Mishnah. By attaching the stool to the trough, what was created is not a seat but a useful shelf for setting down utensils: the one kneading in the trough needs a place on which to rest the water vessel or the container holding the sourdough. The question is whether the stool retains its status as something designated for sitting and for bearing weight, and is therefore susceptible to midras, or whether it has become secondary to the trough and is nothing more than a shelf, and therefore becomes tamei in the way the trough does, but not with midras tumah.
The wording of the Mishnah: "Kisei shekeva'o ba'areivah, Beis Shammai metam'in u'Veis Hillel metaharin" - a chair that one fixed into a kneading trough, Beis Shammai declare it tamei and Beis Hillel declare it tahor.
"Beis Shammai metam'in" - Beis Shammai declare it tamei: here the intent is not merely that the utensil is susceptible to tumah, but that it is susceptible even to midras tumah. The utensil was originally built as a seat, and even after it was fixed into the kneading trough a person is still liable to sit on it. In the times of the Mishnah, when furniture was scarce, someone who had no chair would sit on the shelf of the trough. For that reason the utensil retains its status as a seat and is susceptible to midras tumah, to the point that it becomes an av hatumah when a zav sits on it.
"Beis Hillel metaharin" - Beis Hillel declare it tahor: here too the intent is not that it is entirely insusceptible to tumah, but that it is not susceptible to midras tumah. If a zav or zavah touched it, the shelf becomes tamei - but not as an av hatumah, rather as a rishon l'tumah, on account of contact.
Shammai, the most stringent of them all, says: "Af ha'asuyah bah" - even one made together with it: not only does a stool attached to a kneading trough retain its status as a seat, but even a kneading trough that was built from the outset with a shelf, where that shelf is fit for a person to sit on - since in certain situations people would sit on it - that alone is enough for it to be considered a seat and to be susceptible to midras tumah. Beis Shammai disagree and hold that it is nothing more than a shelf on a kneading trough, and since it is not designated for sitting it is not susceptible to midras tumah, and Beis Hillel certainly agree with that.
In summary: this Mishnah contains two cases. In the first - a bride's chair whose coverings were removed: Beis Shammai declare it tamei because it is still fit for anyone to sit on, Beis Hillel declare it tahor because it has lost its designated purpose, and Shammai is stringent to the point that even the frame of the chair alone is susceptible to tumah; and the halachah follows Beis Hillel. In the second - a chair fixed into a kneading trough: the discussion concerns susceptibility to midras tumah, which is unique to things meant to bear a person's weight; Beis Shammai declare it susceptible even to midras, Beis Hillel exempt it from midras and hold it liable only to tumah through contact, and Shammai is stringent even regarding a trough in which a shelf was made from the outset.