I do not purport to deduce them from axioms but from two things:
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Scarcity
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Self-ownership
Before we get any further, I propose that the term “private property” is of superficial use - “property” by itself means “allowing” one party to use a certain resource while denying that power to another party. “Private” is more adequately replaced by “individual.”
Furthermore, keep in mind that I’m trying to establish a rational basis for what is generally called “natural rights”, more adequately described as “legitimate interactions in society”, and I’ll leave the term “legitimate” for you to define (I realize this might be no easy task).
Once this system of natural rights is established, you must follow this rule to remain legitimate:
It is illegitimate to violate the rights of a “peaceful man” (where peaceful is defined as a man acting within his rights, which shall be established later).
On to the deduction.
I’ll start with (2) - self-ownership. This is not an axiom. This is a proposition. Yet I contend that it is a necessary proposition for establishing any sort of society. If you do not agree that we each have self-ownership, we really have to further use discussing the matter and we will live in a state of no established respect for each other.
I do not believe it a very onerous proposition.
So now we have
It is illegitimate to violate one’s self-ownership and any other rights deduced hereafter.
So we have agreed to self-ownership.
Now, for a brief discussion on (1) - scarcity.
We live in a world of scarcity. This means that goods are not all unlimited in amount. In the Garden of Eden (or at least a fictional garden), you can extend your arm and pull whatever you want out of thin air. You can wish resources into existence with no cost to you infinitely. When resources are non-scarce, there are no conflicts between people over resources because even if I take your Mustang, you can wish another one into existence anywhere and you have not been prevented from using the good directly. In general, if I “take” a car from you, you still have 100% use of the car anytime you like. There is no need for property. Theft is inconsequential.
Yet even in the GoE, there are some scarce resources. One is your body. The other is the space in which you stand.
Moving out of the GoE into the real world of scarcity, we have a limited amount of goods. If I steal your car, you can no longer use that car. There is a 0-sum game of physical resources going on.
When we have scarcity, people are competing for a finite amount of goods. When two people want to use the same good at the same time you have a problem. I can’t be swinging the same axe as you at the same time. I will be physically messing with your body, thus violating your self-ownership.
Hence, we see that a conflict arises. We’d like to establish some rules for resolving this conflict so that self-ownership is not violated. There has to be one person who is legitimately using the resource and another who is not. That, my friends, is the definition of property - the exclusion of others from a resource.
And I have hence deduced property from scarcity and self-ownership. We see that indeed property is an extension of self-ownership.
Now, for another point regarding property - can groups own property collectively? No. For more info, I point you to Walter Block’s responses to Caplan on FracR and joint ownership in marriage - http://www.lewrockwell.com/block/block110.html - to summarize: Due to methodological individualism (the only thing that makes sense, really), you cannot have two people at the same time have 100% ownership of something because you again have a conflict of self-owernship if they both try to use it at the same time. We cannot both be 100% legitimate when I try to use $5 to buy a soda and you use the same $5 to kindle your fire. Hence, property must be individual and not collective.
I do not claim to have deduced the “right-libertarian” property - from homesteading until abandonment or death do you apart. I just claim to have deduced the fact that property is a necessary deduction from self-ownership if you want logical consistency.
Indeed, if you think about it, literally all ideologies I’ve heard of advocate property in some form - whether it be the individual “permanent” property we want or the collective property that AnComs want. We all advocate a monopoly over a resource - the left cannot get away with claiming that we’re the only ones.
So there you have it. Please critique.