The proof breaks down here: "If this were the case, we would expect everything to be perfectly communal."
The word "communal" is ambiguous. I take "communal" to mean either "being used by everyone all the time" or "everyone has the right to use it at any time".
If you're using the second definition, then basically the statement is saying "if there were no rights, then these rights exist". That might be the conclusion, but the right-hand side of that statement does not follow from the left-hand side.
If you're using the first definition, then the statement is saying "if there were no rights, then everything would be used by everyone all the time". However, I do not think this would be logically necessary. It strikes me that if there were no rights, people would use things as they see fit, taking into account the possible negative side-effects of using things others intend to use. I wouldn't say this is communal use.
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