Whats better about using the "deterrent" excuse compared to the "revenge" truth? Why would it even deter a potential death-row criminal? How many future death-r

Whats better about using the "deterrent" excuse compared to the "revenge" truth? Why would it even deter a potential death-row criminal? How many future death-row criminals would have not committed atrocities because of that particular death? Whats the big deterrent about dying compared to the alternative: a life in horrible prison system where you may just die a more unpleasant death anyway?