On the surface, the argument makes sense.
Rational people understand links between cause and effect and crime and punishment.
A fear of death or the possibility of death also affects the behavior of most reasonable people. People who murder, however, are rarely rational at the time they commit the crime. Hired killers obviously assume they will not be apprehended. The death penalty never has shown to benefit a society. In fact, there are strong indications that it increases people's tolerance of and tendency toward violence.