Living is the Question. Is 'Still breathing' enough for living? If the punish was say . . . Paralyzation from the Neck down? Waiste Down? would that leave your sensabilities unflettered Rocket River
I'm not sure how life imprisonment is not a fulfillment of vengeance. Maybe it's not as complete as the death penalty, but it has to get you 95% of the way there. Besides that, I don't know if one should list vengeance as a legitimate purpose of punishment. It certainly would be a thing again from justice, which you neglected to list. As for justice, I don't think you can always say that life in prison is just as appropriate as the death penalty. It would depend on the crime. Life in prison is not as effective in incapacitation as the death penalty (except insofar as it takes decades to actually execute someone). The prisoner is unlikely to rape more children, but can commit other crimes in prison. Once a man is executed, he can't commit any crimes at all.
That's not my "reasons for punishment". Those are taught in any Government class that gets into theory. Justice is too broad a term, it means to much to be a reason for punishment.