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As far as justice...I got your justice right here. A balance in one hand to weigh the sins...and a sword in the other to deal out the punishment..and they call her the statue of justice. 
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...and we all know how far justice goes since the creation of this statue...
It would pretty much suck to be one of the about 5 people who are murdered each year by the US government yet are later found to be innocent, no?
Someone who has committed one or more gruesome deeds does not belong in a society, so far everyone here can agree, I hope. However, once this has been determined, what does one do? What does one decide?
One can let anger and ignorance influence the judgement and sentence this person to death, or to stay for a long period in a building which is supposed to keep 'criminals', as they're called', away from society.
But does this solve the 'problem'? No, of course not. By murdering the murderer, one accomplishes nothing, it only shows the citizens of the country that the government condones murder in certain cases, making murder less 'wrong'.
A jail-sentence accomplishes very little as well, because by maltreatment of a 'criminal', he or she won't suddenly change into a 'good citizen'. It's more likely that they'll forever hold a grudge against their country, making them even more dangerous.
The only just course of action is to investigate why they commit these 'crimes' and what can be done about it.
But the general public doesn't give a damn about 'justice' and what's just and what's not. They want to see blood. Vendetta. Vengeance. I truly wonder what makes us better than the citizens of the Roman Empire who enjoyed the slaughtering of wild animals, criminals, Christians and gladiators, who were forced to fight each other for the 'glory' of the Empire.