You make your first mistake in claiming that law requires government. It does not.
Okay, I don't follow this step. If me and my community agree that some hypothetical judge is capable of resolving our disputes and that his decisions are binding, then we've just made him into (part of) our local government. That is what gives him the authority to determine the law; we have agreed that he governs us in the matter.
Whereas, if me and my community do not agree to abide by the will of Judge Hypothetical, meaning that there is no communal government, then he has no power and cannot effectively determine the law.
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I don't like this duchy. Now, it's an adventurer.