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The difference is that the retributive argument is based on what the criminal deserves and the utilitarian argument is based on what is best for the society as a whole.
A utilitarian argument for the death penalty would be based on the idea that it will deter crime and make society as a whole happier than it would be without the death penalty. The death penalty detracts from the happiness of some, but it increases overall happiness by deterring other people from committing crimes.
Notice, though, that this argument is silent on the issue of justice or of what the criminal deserves. The retributive argument is not silent on this issue. It is based wholely on the idea that crimes and criminals deserve to be punished. This argument does not address the future. Instead, it deals with the past. It says that what the criminal has done is such a bad thing that he or she deserves to be killed.
Retributive theory looks backward and tries to determine what the criminal deserves. Someone who does something that is bad enough should be killed. The utilitarian theory looks ahead at the consequences of the punishment. The death penalty is good because it will deter potential criminals and bring greater happiness to us in the future.
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