No, also, the three unconditioned dharmas also lack inherent existence since they also lack a self.
There are two kinds of space discussed in Buddhist texts. The first and most important is space as "absence of obsctruction." This is uncompounded or unconditioned space. The second kind of space is dimension, such as the dimension of the cavity in a cup. That kind of space is compounded.And no, I have no idea how uncompoundedness could be proven by logic.
The other two unconditioned dharmas, the two cessations, also lack inherent existence because they are the absence of causes, and do not by themselves exist.