We begin the fourth chapter of Maseches Eduyos, chapter 4, mishnah 1. The theme of the chapter is announced in its very opening words: "Eilu devarim mikulei Beis Shammai umechumrei Beis Hillel" - these are matters among the leniencies of Beis Shammai and the stringencies of Beis Hillel. The mishnayos before us list the disputes in which the stringent opinion belongs to Beis Hillel and the lenient one to Beis Shammai - something fairly rare, since Beis Shammai are usually the stringent ones, and that is precisely why these exceptional disputes were gathered here. This is a famous chapter, collecting a wide variety of subjects from halachah to aggadah, and in all of these cases the halachah follows Beis Hillel.
An egg laid on Yom Tov:
The first case is an egg laid on Yom Tov, which is also the opening mishnah of Maseches Beitzah - one of the most complicated topics in all of Seder Moed. For the full treatment one should study that mishnah there; here we will suffice with the essentials.
Beis Hillel hold that there is an independent Torah law known as 'hachanah d'Rabbah'. Its principle is this: one may not use on Shabbos or Yom Tov a food that was not prepared on the preceding day, on a weekday that is neither Shabbos nor Yom Tov; the preparation must take place on an ordinary day. The source of the law is the verse stated regarding the gathering of the manna: "And it shall be on the sixth day, and they shall prepare what they bring" - on the sixth day, which is an ordinary weekday, one must prepare what will be brought. This is preparation in advance.
The basis of the law is that even food prepared by Heaven, that is, through natural forces and without any human involvement, requires advance preparation on a weekday and not on Shabbos or Yom Tov. Hence the problem with an egg according to Beis Hillel: if the day preceding Yom Tov was Shabbos, it turns out that the hen, and in effect the forces of nature, 'prepared' the egg on Shabbos and not on a weekday, and therefore an egg laid on Yom Tov is forbidden.
This is not a law of muktzeh in itself, nor is it a matter of nolad, but a separate law of preparing food from the day before. True, once the egg is forbidden to eat it becomes useless and at that point it also becomes muktzeh, but the muktzeh is the result of the prohibition and not its cause.
This law applies equally on Shabbos and on Yom Tov, except that on Shabbos one does not eat a raw egg anyway. Seemingly it should not apply to a Yom Tov that falls after a weekday: if Shavuos falls on Tuesday, then Monday was a full weekday, and an egg laid on Shavuos should be permitted even with these restrictions. However, the Sages forbade it by rabbinic decree, lest people come to eat even an egg laid on a Yom Tov that falls right after Shabbos.
In the language of the mishnah: "Beitzah shenoldah b'Yom Tov" - an egg laid on Yom Tov - "Beis Shammai omrim tei'acheil" - Beis Shammai say it may be eaten, since they do not hold of 'hachanah d'Rabbah', and they do not restrict food prepared by natural forces on the Shabbos preceding Yom Tov. "U'Beis Hillel omrim lo tei'acheil" - and Beis Hillel say it may not be eaten, because they do hold of 'hachanah d'Rabbah', and the egg must be prepared on a weekday and not on the Shabbos preceding Yom Tov. And as stated, even on a Yom Tov that does not adjoin Shabbos it was forbidden by rabbinic decree.
The measures of se'or and chametz regarding the prohibitions of bal yeira'eh and bal yimatzei:
The second topic in our mishnah concerns the prohibitions of bal yeira'eh and bal yimatzei on Pesach. Simply put, there are two categories of things forbidden on Pesach:
Chametz: a food made from grain that has leavened - from the five species of grain (wheat, barley, spelt, oats and rye) that came into contact with water and was left for eighteen minutes without being worked, until it rose and leavened.
Se'or: the fermenting agent used to leaven dough. Today we add ready-made yeast, whereas in the time of the mishnah they would take dough or a loaf of bread from the previous day that had already spoiled and in which the yeast had fermented, and add a bit of it to the fresh dough in order to leaven it. The se'or itself is not a food - it is like spoiled bread that one does not eat - and nevertheless it too is forbidden on Pesach.
And corresponding to them there are two categories of prohibition:
The prohibition of eating: One may not eat chametz or se'or on Pesach.
The prohibition of owning: One may not even keep them in his possession. The Torah states two separate prohibitions in this regard - bal yeira'eh and bal yimatzei, that a person may not see it and it may not be found in his possession - and for our purposes we will render both of them as a single prohibition: one may not own chametz or se'or.
The wording of the verse regarding the prohibition of owning is doubled: "And no chametz shall be seen by you, and no se'or shall be seen by you throughout all your borders." Why is "and it shall not be seen by you" repeated? Why does the verse not say briefly, "chametz and se'or shall not be seen by you"? What does the repetition add?
Beis Shammai learn from here two separate prohibitions, since two different measures apply to them. Se'or, whose leavening power is more potent, is forbidden to own even in the smallest measure that applies to eating foods - a kezayis, the volume of an olive. Chametz, however, which is an ordinary food, has a larger measure - a koseves, which is a date. Generally a kezayis is the smallest measure, and above it is a kebeitzah, the volume of an egg, while the koseves is an intermediate measure between the two.
It emerges that Beis Shammai are the lenient ones here: even though one may not own even a kezayis of se'or, the measure of chametz that one is forbidden to own is a koseves. With regard to eating, all agree that the measure for both is a kezayis, for even though se'or is not a food, one who eats it thereby treats it as a food ('achshevei') and it becomes a food as far as he is concerned. The dispute, then, concerns only the measure for ownership: Beis Shammai are lenient and say a koseves, and Beis Hillel are stringent and say a kezayis.
And in the language of the Mishnah: "Beis Shammai omrim se'or bekezayis vechametz bekoseves" - the measure of se'or for ownership is a kezayis, and the measure of chametz, which is larger, is a koseves. "Beis Hillel omrim zeh vazeh bekezayis" - the measure of both of them, se'or and chametz alike, is a kezayis, and according to all opinions the measure for eating is a kezayis.
In summary: This Mishnah lists two cases in which Beis Shammai are lenient and Beis Hillel are stringent: an egg laid on Yom Tov, which according to Beis Hillel is forbidden because of the principle of 'hachanah' taught by Rabbah, and the measure for owning se'or and chametz on Pesach. The halachah in both cases follows Beis Hillel: the measure for owning chametz, like that of se'or, is a kezayis, and an egg laid on Yom Tov may not be eaten at all.