In the UK you can do something unless they pass a law against it, in a positivistic legal system if the government ries to stop you doing something you have to prove a legal right to be able to do it. In practice it does work like that, just like the US government has to justify its rights of government (the opposite principle).
Also, positivism assumes that Rights are not inalienable, worse (as you have already noted) it decouples Justice and Law. Justice is not "man-made" any more than "happiness", Justice is a metaphysical concept. A legal system without metaphysical undergurding is just due process.
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