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Read this article by the aclu (american civil liberties union) http://www.aclu.org/blog/capital-punishment-human-rights/right-life-denied-death-penalty-violates-constitution-and
Eighth Amendment, which prohibits "cruel and unusual punishment." The tricky bit is that capital punishment was certainly not viewed by the Founders as "cruel and unusual," since people were routinely hanged for crimes as minor as theft in those days. They meant things like breaking on the wheel or drawing and quartering. So you need to make a "living Constitution" argument that the Founders meant for the *specific* punishments prohibited to depend on current ethical standards, whatever they might be. The problem with *this* is that it's a two-edged sword. Sure, you can argue we are more civilized now, and we don't like putting people to death even for murder, so the Eighth Amendment has "evolved" to mean the death penalty is unconstitutioal. Now think this out further: what happens if society changes again, and in 50 years' time citizens generally feel like breaking on the wheel and disembowelment are no longer "cruel" or "unusual." Then we should say the Eighth Amendment has "evolved" again, and now allows those much harsher punishments? If the Eighten Amendment evolves all the time, according to the public ethical standards of the moment, then it pretty much means nothing. It just means "don't impose punishments you find distasteful" -- and of course we don't need a Constitutional Amendment to tell us to do that.