In An Introduction to the Principles of Morals and Legislation Bentham states that motivation is key to understanding law-abiding behaviour and punishment as a result. Punishment should be weighted so as to change people's motives. Mischief could be divided into two parts - 'primary' mischief which related to the pain sustained by an assignable individual. 'Secondary' mischief extended throughout wider society affecting innumberable unknown, unassignable individuals. Bentham argues that punishment should also be proportional - the less harm from the offence, the less pain in the punishment.
Punishment is a bad thing in utilitarianism because it brings about displeasure - it can only be justified if it prevents future harm. Bentham's work claims that punishment cannot be adminstered in the name of retribution.
Bentham and Singer are against capital punishment but Mill is for it - claiming that there is no intrinsic right to human life. All rights serve to maximise happiness.
Utilitarians would have to look for empirical evidence if they ever wanted to make capital punishment legal - does the death penalty stop future…