If someone kills my son by running them over and it would have been morally and legally justified to have shot them before they ran over my son then why can't I simply walk up to them anytime after they killed my son and shoot him in the head? Why can I shoot someone who breaks into my house tonight but I can't go to their house and kill them tomorrow?

What changes between the initial interaction; for example, someone is in a shootout with the police and therefore the police officers are justified in killing the attacker and catching them later in a hotel where the person is taken into custody? Why not shoot them right there?

Is it due to an uncertainty rule? Because at the moment you are certain who to fire at but in the future, you may not need to fire at them. The police, under the guidance of the rule of law, are instructed to inflict the minimum necessary force to enact peace. Also, they have an incentive to de-escalate any interaction. Any time someone is enforcing rules on another person there is a potential for reactive violence. By escalating an interaction an officer increases the likelihood that they will become injured or killed.

Or is it due to a hierarchical rule? As in, the lower level enforcers are only allowed so much power. Whereas the real power to sentence is in the judge's hands. The elite group of, hopefully, intelligent people who have the power to kill. (If the state has a death sentence.)

I believe that in the past the rule of law was much less adhered to, not that it is overly adhered to today. Also, enforcers of the rules would have more leeway and flexibility to enforce and enact justice as they saw fit. They might just hang someone on the spot. Kill them without any words spoken. So long as the sentence was carried out it didn't matter so much how.

Is it a way to appease the masses? By creating a large hierarchical structure that dictates procedures and rules the burden of power is diminished. Having judges, juries, and law enforcement agencies all involved with enacting justice any mistakes can be recognized and corrected as opposed to a dictatorship, where one person is in charge of all acts of justice and therefore is either allowed to continue their reign or they are ousted, one way or another.

If people are allowed to enact justice on their own like vigilantes and a mistake is made the person may never be found, which satisfies no one, or, even if they are caught, we may never know the truth as the other person is dead and can not defend themselves.

Back to, uncertainty, if someone watches someone murdered there is a good chance that within a short period of time they might be able to recognize them as the attacker but the lesson from so many inaccurate eyewitness testimonies is that one can't always trust their own recollections. So the uncertainty principle should be accounted for in the personal case as well.

So how can we determine certainty? What if you are certain that you know the killer's face so you find them and kill them just to find out that the person you killed is the twin of the actual killer. This is a rare case and one would have to be convinced that the killer had a twin and that is plausible that you killed the wrong one. In a normal case, we have evidence, argumentation, and consensus to take on the task of providing relative certainty.

DNA and other forensic sciences have made a great deal of progress in the pursuit of certainty scientists are not perfect. Cameras and other recording devices have done some work here as well even though they can be doctored. Lawyers attempt to plead the case of each party and we trust reason through argumentation to win the day. We trust that judges know the law and will consider the case objectively, without prior judgments or prejudices. (Not perfect either.)

Are juries a good idea? If it is a truly random set of people who end up on a jury then the odds of getting someone with a below average IQ is very high. The odds that the jurors will not have some prejudice against the defendant or prosecutor is slim to none. Most people have a rudimentary understanding of the law at best. Many people have a hard time paying attention. People are vindictive, jealous, overly agreeable, easily confused and distracted by sex. But when a jury convicts someone we can blame them and since they all agreed there is a consensus. They represent the public and act as a kind of intellectual microcosm of the society's "will".

One solution is to mix the jury concept with the judge concept and create a random selection of three judges to oversee all cases above a certain level. Less significant cases would be overseen by a single judge where appeals would allow for second opinions and oversight. In more significant cases, especially violent ones, the three-judge panel would decide the case and appeals would be taken straight to the federal level where three judges would oversee the appeal. For this system to work there would have to be either more judges, less litigation, or a combination of both.

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The system can't be perfect but how do we minimize its flaws?

How many people are we willing to wrongfully punish and how many people will we let go who should be punished? How many people are rightfully accused and rightfully punished? How many are rightfully accused but wrongfully punished? The justice system is a reflection of society's attempts to answer these questions. Throw in some corruption, ignorance, and prejudice and you have the idiosyncratic character of a society's justice system.

The American system seems to have become bloated and mismanaged and has created corruption of certain aspects of the justice system. Namely, too many laws creating too much wasted litigation and punishment. Crimes are invented and rarely discarded but when a law is discarded the real victims are those who were rightfully imprisoned at the time but who are now suffering the consequences of the punishment for a crime that is no longer deemed wrong.