Chapter 1, Mishnah 13 - the third of the four cases in which Beis Hillel retracted their view and ruled like Beis Shammai. This Mishnah appears in Maseches Gittin as well (Chapter 4, Mishnah 5), and it deals with the unusual case of one who is "half a slave and half a free man" - a person half of whom is held as a slave and half of whom is free.
How such a situation comes about:
A man, let us call him Yaakov, buys a slave. The slave becomes his Canaanite slave, immerses in a mikvah, is circumcised, and from then on he too is obligated in mitzvos like a Jewish woman. Yaakov dies and leaves two heirs, Reuven and Shimon, and the two of them become joint owners of the slave - let us call him Eliezer.
For some reason Reuven frees his half, and Eliezer turns out to be half Jewish. After all, a Canaanite slave has already immersed and been circumcised and keeps mitzvos like a Jewish woman, and when he is freed he becomes a free man and a full Jew, something like a convert who enters through the back door. Here, however, only half of his person was freed, while the other half is still a slave owned by Shimon. The result is a person half of whom is obligated in mitzvos like a Jewish man and half of whom is owned and obligated like a Jewish woman - and from here a number of complications are born.
The first problem - the financial problem:
Eliezer's person, and in the most practical terms his earnings, still belong to Shimon. But Shimon faces a difficulty: everything Eliezer does is half for his own benefit and half for Shimon's benefit. If Eliezer refuses to work, Shimon has no recourse, since half of him is free, and the half that is free cannot be forced to work. Jews are free men by nature, and slavery is not imposed on them.
The Sages therefore instituted an arrangement, and this is the starting point of the Mishnah: instead of every penny of output being jointly owned half and half, the days are divided up. On alternating days - say Monday, Wednesday and Friday - Eliezer works for Shimon, and on Sunday, Tuesday and Thursday he works for himself. On the days that belong to the master, Shimon may compel him to work and enjoy all of the output, and on the days that are his own, Eliezer may do as he pleases, as a free man does. This solves Shimon's financial problem.
The second problem - the fundamental problem:
And what about Eliezer himself? Since he is half free and half a slave, there is no one he can marry:
A Canaanite slave - may marry a Canaanite maidservant, and build a family with her.
A free Jew - may marry a free Jewish woman.
But a Canaanite slave does not marry a free woman, and a free Jew does not marry a Canaanite maidservant.
Since this person is half free and half a slave, as though every cell of his body were half free and half a slave, there is no one in the entire population whom he can marry: he cannot marry a Jewish woman because he is half a slave, and he cannot marry a Canaanite maidservant because he is half free. So what is he to do?
This is a truly fundamental problem. Building a family stands at the very purpose of creation, and it is a basic principle of the human experience and of what the Holy One, Blessed be He, created man to do. Yet this person cannot build a family, since he cannot marry. True, there may be other matters in which he is caught in some strange halachic tangle and cannot act, but that is nothing like being prevented from bringing children into the world, which is the very purpose of creation and of the human experience, and it is not the will of the Holy One, Blessed be He, that he be prevented from it.
The Mishnah's ruling:
This brings us to the enactment designed to solve the problem. Beis Shammai say that we compel the master to free his slave so that he will be able to marry, and in the end Beis Hillel agree that this is the better of the two options. Even so, the master should not have to absorb a financial loss, since we are talking about an economic asset that belongs to him. Therefore the freed slave writes a promissory note to his former owner and pays when he is able: if a slave is worth one hundred on the market, he writes a note for fifty - half his value.
The words of the Mishnah:
"Mi shechetzyo eved v'chetzyo ben chorin" - a person who is half slave and half free, whether in the case described above or where two partners bought the slave and one of them freed his share. It is worth noting: even if a hundred people inherited the slave or purchased him and one of them freed his share, since he is partly owned and partly free - that is "half," and the same rules apply whether there are two owners or ninety-nine owners and one who frees his share.
"Oved es rabo yom echad v'es atzmo yom echad" - this is the enactment of the Sages: one day he works for his master and the produce belongs to the master, and on alternating days he works for himself and keeps the earnings for himself. "Divrei Beis Hillel" - this is the position of Beis Hillel.
"Amru lahem Beis Shammai: tikantem es rabo" - you have found a fine solution to the master's financial problem, for now he can compel the slave to work on the days allotted to him. "V'es atzmo lo tikantem" - but you have not solved the essential problem of the half slave himself, that he cannot marry and build a family. "Lisa shifchah eino yachol" - because he is half free. "U'vas chorin eino yachol" - because he is half slave. "Liba'tel" - shall he sit idle and never marry? Such a thing is inconceivable.
"V'halo lo nivra ha'olam ela l'piryah v'rivyah" - the entire purpose of existence is that people build families, and this person is unable to do so. "Shene'emar: lo sohu bra'ah, lasheves yetzarah" - a verse in Yeshayahu: He did not create the world to be a wasteland, but so that it would be settled with people, and from here we learn that a person must see to it that he has children.
Why is the verse from Yeshayahu needed?
Seemingly we have an explicit verse in the Torah, and the mitzvah of "be fruitful and multiply" is even stated there twice. However, that mitzvah applies only to the free half of this person, since in his slave half he is obligated in mitzvos like a woman, and a Jewish woman is not commanded in "be fruitful and multiply." The verse "lasheves yetzarah," by contrast, teaches that every person is meant to take part in settling the world, and this obligation applies both to the slave portion and to the free portion within him. That is why this verse is preferred.
"Ela mipnei tikun ha'olam" - in order to make the world a better place and to address this problem, "kofin es rabo" - we compel the master, "v'osin oso ben chorin" - to make him a free man and a full Jew, so that he will be able to marry and build a family. "V'kosev shtar al chatzi damav" - the freed slave writes a promissory note to his former owner, committing to pay half of his value as a slave when he is able.
"V'chazru Beis Hillel l'horos k'Beis Shammai" - Beis Hillel retracted their position and conceded that the words of Beis Shammai are more compelling.
Although the Torah contains the prohibition of "l'olam bahem ta'avodu," that one may not free slaves for nothing and there is a mitzvah to keep them in servitude - and the reason for this is for the slave's own benefit, since servitude in a Jewish home, keeping kosher and Shabbos and the like, is to his true advantage, for as far as his eternity is concerned he is a soul and not a body - nevertheless it is permitted to free a slave for the sake of a mitzvah, even a rabbinic mitzvah. And here the need is greater than that, and therefore the right thing to do is to free him. This is how Beis Hillel ruled, and this is the halachah in practice.