One of the things that makes the Greco-Roman world seem so alien is the practice of killing unwanted babies. However much we are shocked by the recent discoveries of the remains of 8 new-born babies in France (and whatever human tragedy underlies it), the fact is that such things were common in antiquity. Some were disposed of by 'exposure' (put out on a dung heap), others by rather more aggressive forms of murder. There were many Roman houses where dead babies lay under the floorboards or buried in the garden.
Indeed the very myth of the foundation of Rome itself starts with a story of exposure. Their wicked uncle ordered Romulus and Remus to be disposed of, and it was only good luck that meant that they were washed up on the banks of the flooded Tiber -- to be found by the wolf (the lupa -- a word which in Latin could also mean 'prostitute', which led to all kinds of predictable speculation on the part of hard-headed Roman rationalists).
And several of the plots of Roman comedies feature a slave girl who turns out to be have been born free -- but exposed at birth, and picked up by someone to be brought up as a slave. The revelation of free birth usually means that she gets to marry the (free) boy of her dreams.
But it's not just in myth and fiction. There is plenty of evidence from real life too. A famous papyrus letter from husband to wife in Roman Egypt, asks her to "let the baby live, if it is male; if it is female, expose it'. And the Jewish writer Philo, writes (disapprovingly) about the practice, explaining that some people strangle or suffocate the babies, while others expose them (after which they are just as likely to be eaten by wild animals as rescued by another human being).That's in Special Laws III, 114-5.
Classicists have found this puts them in a moral spot. Some have denied that the practise was, in real life, a common as is often imagined. One ostrich writing in the 1930s (to be precise, Classical Review for 1932) claimed that "the cruelty involved in infanticide even by exposure is very slight". The more recent line has tended to be that we are dealing here with a significantly different definition of "humanity" from our own. Of course, we have plenty enough debates about when someone becomes a person with human rights (at conception? at birth? somewhere in between?). For the Romans it was when the person was formally "accepted" (usually by the father) into the community, at a few days old. Before that the baby, although outside the womb, was no more than a foetus; infanticide, in other words, was a late abortion.
That is all logical enough, though I suspect in a hundred years or so, it will look as desperately exculpatory as the 1930s garbage about it not being 'cruel now does.
But why did the Romans (and the Greeks too, for that matter) do it?