[Faceman]

To clarify,
these two sets of theories I presented are not only about DP but about criminal justice in general.
about prevention:
Special Prevention is trying to keep a perpetrator from striking again by
locking him up, psychological treatment, etc.
In the recent years it has turned out that imprisonement in general makes people even more socially incapable, so in many countries (e.g. Austria) long-term sentences are avoided as good as possible (of course you'll still be in for 20+ years with murder or something comparable, or 10+ years for your umpteenth offense).
General Prevention actually gets more important with really small "crimes", where the consciousness of guilt is not really there. Perfect examples are traffic violations such as speeding or overtaking a traffic jam on the restricted lane. These are things that can be tempting for the otherwise law-abiding citizen too. But if there is an immediate reaction by the police and the guy who was just seconds ago successfully cheating is pulled over, people will be content with their law-abiding ways and won't copy his behaviour.
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Death Penalty does not serve General prevention, as people who commit a murder usually don't factor in how much time they'll have to do for it or if they get executed. Generally harsh punishment doesn't deter many, because people are optimists and don't want to think about getting caught.
And it does only serve special prevention in a very crude and simplicistic manner which completely neglect the second side of special prevention: making the culprit a useful member of society again.
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Oh, and about DP providing closure. Wouldn't locking the guy up for life and never hearing of him again provide closure too?
I figure that quite a few of the guys on death row are there because they thought that murder was the only way to get closure with their ex-girlfriend.