The purpose of the legal system is not vengeance, its justice
Justice is morality concept, so:
Let's look at the death penalty from a moral standpoint
You state that killing people is wrong, so you punish people who kill
That's how the law works in regards to murder
It's a rule system of morality, the rule being "Do not murder" with murder defined as killing with intent (intent to harm, for second degree murder). When you execute someone, you are killing them, and you have intent to kill them.
You are breaking your own rule, which means your structure you define as justice is not universal, and therefore bad

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"Whenever a man does a thoroughly stupid thing, it is always from the noblest motives" (Oscar Wilde).
"Patriotism is the virtue of the vicious" (Oscar Wilde).