I know these things should be in order, but
7. "We" are that which is capable of thought.
3. If we think, then we have that which is capable of thought (the mind).
4. if we have that which is capable of thought, then it must be supported by another corporeal mechanism (the body).
Not all of the body is necessary for supporting thought. You don't need hands to survive and support your thinking mechanism, for example. Does that mean we don't own our hands? They aren't part of the relative matter of that which is capable of thought.
Our hands are connected to the relative matter of that which is capable of thought. Going back to our master-slave relationship, being a master of a slave entitles you to that slave as well as all that is produced by that slave; you could make your slave mix their labour with some land and that land would be yours, not theirs, as they are property: they cannot own.
5. The relative matter of that which is capable of thought is the corporeal mechanism.
6. If there is no corporeal mechanism, then we are not capable of thought (therefore we are not).
7. "We" are that which is capable of thought.
8. The corporeal mechanism upon which we are supported is therefore naturally belonging to us.
But that's not what "belonging" means. Being a constituent of something doesn't mean it owns you.
One, that is one that is capable of thought (and therefore can own), cannot be owned. Something that cannot think being a constituent of that which can think entitles that which can think to its constituent parts. The cells that make me belong to me. The organelle that constitute a cell do not belong to the cell; they belong to me.
9. If the corporeal mechanism upon which we are supported is therefore naturally belonging to us, we own our bodies.
10. Owning something is owning the rights to something, which effectively makes that something an extension of our own rights and of ourselves.
"Owning" something is only owning the rights to something under certain artificial institutions of owning.
Could you give me an alternative institution of owning?


