Edwin Judge clarifies theory and practice when it comes to newborn children in Greece and Rome.
Infanticide was ethically permissible only in the sense that every father in Rome had to make the decision whether to accept a child when born or not. The father had to pick him up, literally. A father might decide not to pick the child up. The child would not be killed, but would be picked up by somebody else – either somebody else in the household or simply deposited on a rubbish heap, in which case a stranger would pick him up, very probably, and rear him in servitude in his house. So I don’t think it’s the case that Romans actually killed newborn children. They don’t practice infanticide in that sense.
The theoreticians, however, in classical Greece – Plato, for example, argued that we must dispose of surplus children. He authorised, if you like, infanticide. He doesn’t say how to do it and he’s only talking theoretically, but he had high principles and one of them was that only the best people should be born, and others are a kind of flaw in the system and should be got rid of. But I’m not aware that any Romans actually killed children.