Mishnah 5 in the fourth chapter of Maseches Gittin is one of the mishnayos dealing with enactments instituted mipnei tikkun ha'olam, for the betterment of society. This mishnah addresses the case of "mi shechetzyo eved vechetzyo ben chorin" - a person who is half freed and half still enslaved.
How does such a situation come about? A slave is owned by two partners, and one of the partners frees his share: fifty percent of him has gone free, while the remaining fifty percent is still enslaved. The question is how to deal with this situation.
The View of Beis Hillel:
The mishnah states: "oved es rabbo yom echad ve'es atzmo yom echad - divrei Beis Hillel" - he divides his time: one day he works for his remaining master, and the next day he works for himself.
The Argument of Beis Shammai:
"amru lahem Beis Shammai: tikantem es rabbo ve'oso lo sikantem" - by dividing up his time you have taken care of the master who still owns a share in him, since this person belongs to himself half the time and to his master half the time. But you have not taken care of the man himself, and the problem remains fully in place with regard to marriage:
"lisa shifchah - i efshar, shekvar chetzyo ben chorin" - he cannot marry a Canaanite maidservant, a non-Jewish woman enslaved to a Jew, since half of him is already free, and a free man may not marry a maidservant.
"bas chorin - i efshar, shekvar chetzyo eved" - neither can he marry an ordinary free Jewish woman, since half of him is a slave, and an ordinary Jewish woman is not permitted to marry a slave.
"yibatel" - perhaps you will say that he should simply remain without marriage and without children? To this the mishnah responds: "velo nivra ha'olam ela lipriyah verivyah" - the very purpose of the world's creation is to be fruitful and multiply, "shene'emar: lo sohu vara lasheves yetzarah" - the Holy One, Blessed is He, did not create the world to be empty; He formed it to be settled.
The Solution Instituted for the Betterment of Society:
"ela mipnei tikkun ha'olam kofin es rabbo ve'oseh oso ben chorin, vechosev shtar al chatzi damav" - for the betterment of society and for the good of the community, we compel his remaining master to free him, and the slave, in exchange for his freedom, writes a promissory note for half his value, that half which still belonged to the master, and ultimately he pays him the money.
"vechazru Beis Hillel lehoros kedivrei Beis Shammai" - Beis Hillel, who had originally said that all that need be done is to divide up his working time, retracted and ruled like Beis Shammai: he must be freed so that he will be able to marry a Jewish woman, for as long as he remains half a slave he cannot marry anyone at all.
In summary: In this mishnah we learned about a person who is half slave and half free - a situation that arises when one of two partners in the ownership of a slave frees his share. Beis Hillel initially held that dividing his working time is sufficient, and Beis Shammai demonstrated that this takes care of the master but not of the man himself, since he is left unable to marry anyone at all - neither a maidservant nor a free woman - and the nullification of being fruitful and multiplying stands against the very purpose of the world's creation. Therefore, for the betterment of society, we compel his master to free him and the slave writes a promissory note for half his value, and Beis Hillel retracted and ruled in accordance with the words of Beis Shammai.