Doron,
This "parallel" quantity vs. "serial" quantity ...
Help me.
Parallel:
"That's Jack" + "Yes, that's Jack" + "Of course it's Jack + Jack ...
Same id repeated.
Serial:
"I see Billy + Margaret + Sally + Jack."
Different ids.
Serial quantity: the sum of Billy, Margaret, Sally, and Jack is 4.
Is there a "parallel" quantity?
If so what is the "parallel quantity" in the example above?
A bowl of oranges:
Here's an orange + another orange + another + orange again.
Not the self same orange but repeated orange id.
Is this parallel or serial?
Serial, I count (obtain quantity of) 4 oranges?
Parallel,
Is there a quantity? One orange? 1 class called "orange?"
Apparently I was missing whatever your core idea when I thought that "id" meant a class collector, and when I thought that "id" meant a unique individual identity.
Perhaps if you can tell me how we obtain a quantity or sum ... ?
Uh-oh!
The proverbial chair is snatched from under me again.
You said:
Singular comparison like (A = A), (A ≠ A), (~A = ~A), (~A ≠ ~A) is fundamentally different than Non singular comparison like (A ≠ ~A), (A = ~A), and it does not matter if these expressions are True or False.
The considered framework is based on both serial and parallel observations, which are based on both singular and non singular comparisons.
Under serial observation (focused on Singular comparisons) we get anti-symmetric collection of certain ids.
Under parallel observation (focused on Non singular comparisons) we get symmetric collection of uncertain ids.
So Jack + Good Old Jack again is a "singular comparison." of a single id. That's a "serial observation."
But Sally + Jack is a "non-singular comparison" of more than one id. And that's a "parallel observation."
Quantity is obtained under the parallel observation.
Under "serial observation," Jack = Jack = Jack. It's just Jack + Jack again. All you get is Jack.
(but I thought that was the parallel observation of a single repeated id.)
The serial quantity of Jack is ?
Let's see. Maybe this is the meaning:
"non-singular comparisons" gather together elements that don't have a single common class id.
"singular comparisons" gather together elements that have a single common class id.
(individual, unique identity is not meant in connection to "parallel observation.")
So apple + orange + banana is a non-singular comparison.
And "fruit" as a common counting class is irrelevant to this parallel observation.
Ah, that "fruit" as a common classifier is ignored in counting these seprate items,
and more fundamentally that "items (being counted)" must be ignored as a common classifier upon which to obtain quantity in order to have a strictly "parallel" observation,
is a very tricky place to try to stand.
Trying to obtain quantity outside a collection based on a common class identity, I don't see how that's done.