[THIS SECTION NEEDS WORKING but I'm just putting things where they belong in a very quick pass so that I can get this book ORGANIZED.]

*For any given object*, the intersection *and* the union of its
identity set with itself is the identity set of the object, and the
intersection of the identity groups of two *different* objects is
the empty set. This last line defines, in fact, the very essence of
what we mean by ``different'' just as the first two lines encapsulate
what we mean by ``the same''.

Let us relate these two statements back to the Laws of Thought. In an
existential -set theory, is not a set. Our human minds try
to *interpret* it as ``the set of things that do not exist'' within
the existential set Universe in question, but of course no such set *exists* within that set Universe (including the empty set) - it is a
statement even in English. Nevertheless, we recognize the first
of these two relations as the *Law of Contradiction* where as usual,
can be any object or the ``empty object'' corresponding to the empty
set drawn from *within* the set Universe. Rendered in English, it
says that ``the intersection of any object drawn from our Universal set
and not-an-set is not-a-set'' (within the existential Universal set of
the theory). Nor can we form the union of any set object to a
``-thing'' that is not, in fact a set.

In this formulation, the two -statements become requirements of
*consistency* of a set theory. Once one defines a set Universe with
its implicit existential subsets, then any sort of algebraic operation
or set specification that yields a result that isn't one of these
subsets must be , an *inconsistent* result. That isn't a
disaster, but it does mean that this and any subsequent operations
involving that result are also - meaningless - within the
specified theory.