When I wrote it I toyed with qualifying the sentence with "at first sight we may expect..."
I used the word "communal" in accordance with the second definition you stated - "everyone has the right to use it at any time" - but I would replace "at any time" with "equally" - to be more general. The point I was making is that it is impossible to define "communal" without relying on the concept of "just claim," since it would be nonsensical to assume every use would be "unjust" - that is to say that every time someone used an item it would be unjust to everyone else. And, if it was unjust to everyone else, such a notion would imply that that the group - minus the individual using the item - had a "just claim." A world without the concept of "just claim" cannot exist.
Further, "communal property" cannot exist - unless something is in absolute abundance. For instance, air could be considered a virtual communal property. Anything in limited supply cannot be considered communal property since if everyone had an equally just claim to the property (like I said above), anytime someone used the property it would be in violation of the rest of the group's just claims. The smaller the supply gets the less communal an item can be approximated. One important example is an individual. Since individuals are unique, it is impossible for them to be communal.
Assuming that "just" and "unjust" do not exist is also nonsensical. These terms become defined naturally and almost instantly. Even if the words did not exist the concepts still would and would very quickly be given terms. Once a baby bird learning to fly fell into my garden. As I approached the garden, there must have been ten birds (of different species, nonetheless) that began frantically chirping at me - in essence, begging me not to mess with the baby. They may not have a word for "just," but they seemed very familiar with the concept. The concepts of "just" and "unjust" are natural.
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