The ancient mindset would say that they would resemble the dishonorable characteristics of their parents. However, if they didn't they could be freed or in some cases buy their own freedom.
I'm not whitewashing slavery. I'm just indifferent to the plight of people who are all dead now anyway. I also recognize that we would have slavery today if it made economic sense. The two things that helped break the inflexible class systems of the past are fossil fuels and capitalism.
ἡ δὲ κἀκ τριῶν τρυπημάτων ἐργαζομένη ἐνεκάλει τῇ φύσει, δυσφορουμένη, ὅτι δὴ μὴ καὶ τοὺς τιτθοὺς αὐτῇ εὐρύτερον ἢ νῦν εἰσι τρυπώη, ὅπως καὶ ἄλλην ἐνταῦθα μίξιν ἐπιτεχνᾶσθαι δυνατὴ εἴη. – Procopius
Ummaka qinnassa nīk!
*MySmiley*