My argument is very simple. In almost all the nations where the death penalty was recently abolished, we have witnessed a sharp upswing in the crime rate. Therefore, IMO, the death penalty must be there to deter these sort of crimes.
I don't think it's the death penalty per se that's the deterrent. I think it's more the prevailing attitude of particular societies and their lawmakers. If they have something like the death penalty, they are perceived as tough and no-nonsense. If they don't have it, they look passive.