That's not good logic.
"In order for there to be nothing, there must be a distinction."
No. If there is nothing, then that's all there is. It's called the empty set. The quantity of all things that exist is zero.
"The fundamental principle of existence is the distinction."
Distinction is a concept, and regards the fact that individual things that exist can be differentiated. Distinguiability occurs when the first thing other than nothing exists.
That's not good logic.
It is a contradiction to say all of the following:
If you can show otherwise, I and the world of logicians would really like to see that proof.
<sourcery>In order for there to be nothing, there must be a distinction.</sourcery>
No. If there is nothing, then that's all there is. It's called the empty set. The quantity of all things that exist is zero.
In order for there to be an empty set, it must exist. The empty set is distinct form nothing. Therefore, nothing can only exist locally (relative to some context,) and cannot exist universally.
<sourcery>The fundamental principle of existence is the distinction.</sourcery>
Distinction is a concept, and regards the fact that individual things that exist can be differentiated. Distinguiability occurs when the first thing other than nothing exists.
Nothing is also a concept, as are 'exist,' 'fact', 'proof,' and 'thing.'
Distinction does not exist as a side effect of the existence of different things. Different things exist as a side effect of distinction/differentiation (or alternatively, both things and distinctions are co-causitive.)