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You talk as if "authority" is a good thing, or something that protects good people from evil people. It has done the exact and precise opposite for literally thousands of years. "Private" just means you're not FORCED by a ruling class to pay for things you don't want. So you're essentially arguing that you should be FORCED to pay the salaries of abusive, power-happy thugs with badges, instead of being allowed to choose for yourself.

Private sector law enforcement is totally different. I have been doing it for a decade. I made nine arrests, all for violent crimes. I have zero use of force in this time. If you cannot afford it, you have to simply fight better. Without the state to make "laws", you are subject to "rules." This is actually how things work: You cannot break a Law. You can only break rules. Try breaking the law of gravity....