I politely beg to differ.

Capital punishment is most often justified as a deterent to the commiting of murder and similar crimes. As such, this necessarily means that fear of receiving said punishment is the operative means of such deterence.

[NOTE: Whether or not capital punishment actually IS an effective deterent to such crimes is a different discussion, one that we should probably not go into here, since it will likely lead to Political Debate.]

There is also - at least in the past if not currently - the dimension that the public execution of rebels and dissidents was a means of instilling fear of the existing government's retribution into the minds of *prospective* rebels/dissidents.