I don’t consider the state killing somebody in cold blood to be justice.
You might not consider it to be justice, but it is clear this is what the church has always taught.
I think what you are talking about is revenge or retribution.
Given that retribution is in fact the primary objective of all punishment, that would be true. As for vengeance, Aquinas defined it as “
the infliction of a penal evil on one who has sinned.” Given that God himself said “
vengeance is mine” perhaps we should view it a little more circumspectly.
“ought to” - according to who? The Church teaches differently. It teaches that the primary interest of penal sanctions are “rehabilitation and social reintegration of the criminal”.
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The primary scope of the penalty is to redress the disorder caused by the offense.
First, I don’t think you properly understand what retribution is.
Retribution: A penalty or reward that a person deserves for moral conduct. (Catholic Dictionary)
The key point is that it is what a person deserves. Sin deserves punishment; if it didn’t there would be no justification at all that would permit it. In any event,
"redressing disorder" has nothing to do with rehabilitation. The order disturbed by sin is the juridic order; it is an offense against, God, society, and the individual.
An eye for an eye is just. But is it merciful?
The argument from mercy doesn’t really work that well. First, to argue that a life sentence is the merciful punishment is to admit that death is the just punishment, otherwise not executing a person would be the just punishment, not the merciful one. Second, mercy, even God’s mercy, is not universal - not everyone receives it. His mercy requires repentance, so what is the argument against applying the just punishment (death) in those cases where mercy is not appropriate?