This is often written in the following dumb format:
Not (A and not A)
This takes a simple statement in English, inverts it, and then expresses it in math form. Nice way to complicate the issue for people who are not math informed.
This works great with Americans whose schooling has somehow taught them, “I’m no good at math.”
How an entire nation of people can be programmed into that five word sentence by so many teachers spending eight hours a day five days a week for twelve years would be easier to understand if most math weren’t relatively simple and if the job of teachers weren’t expected to educate the public.
Of course the same teacher will quickly point out that the above sentence fails grammatically even though its communication is clear.
Someday I will write a Map Thinker’s Guide to Grammar™. But not today.
So let’s make it simple English and simple math.
Either A exists or A does not exist.
It cannot be both at once.
There is a problem with this.
The Law of the Excluded Middle, which we will deal with next makes that statement.
Either (A or not A)
They make the same basic statement but they mean different things. That is why they are expressed differently. Seen as neither one makes much sense to a Map Thinker™ it doesn’t really matter but we will go over the distinction just because.
Not (A or Not A) means that a thing cannot both exist and not exist at the same time in the same way. Or you can say a statement cannot be both true and false at the same time.
I used the word “way” while most translations I have read say “respect”. I doubt if either one carries the exact meaning of the original. Even if you speak modern Greek I am not convinced you would be certain exactly what the same word meant 2,300 years ago.
Today we know, from quantum physics that some pretty large particles can in fact exist in more than one place at the same time in the same way. The largest I have heard of is a drumstick. The kind made of wood you play a drum with, not the kind from chicken you eat.
But lets take something larger.
Two things I love. Water and Trees.
A tree on a knoll either exists there or it does not. It (at present knowledge) cannot both be there and not be there at the same time. It may not be there tomorrow, it may be cut down, or zapped by lightning.
Unless of course you subscribe to the Quantum concept of parallel universes in which case there will be a universe where the tree is on the knoll and another where it does not exist there. How about the one where the knoll itself does not exist there.
But lets call that reaching.
The question arises, “Does the tree exist at all?”
In point of reality it does not.
The tree is a product of Emergence. That is the tree is built of simple building blocks that are built of even simpler building blocks that are composed of non-particles that are also non-waves.
Ready?
That may sound confusing.
It is.
It is a combination of my understanding of particle physics and complexity theory.
The tree itself is an expression of a combination of relationships that exist in reality in such a way as to produce a species we call “tree” and this is a particular member of that species.
So the tree is really a figment of our imaginations. We aren’t completely sure what is out there. We are sure that this particular combination of universal building blocks is identifiable as both a species and an individual and we call them trees.
So no matter how you look at a tree using modern knowledge it both exists and does not exist at the same time.
And we are talking about something we can cut down to build houses, handcuff ourselves to so others cannot cut them down to build houses, climb, kick, or hug.
In reality the tree does not really exist.
But we can treat it as though it exists in the same way as Aristotle conceived of it.
Things do not get easier when we tackle concepts instead of objects we can touch, hear, taste, hold, smell, see, kiss, or rub on.
A thing can only be true or false if it is narrowly defined. If the parameters are sufficiently delineated. The problem is that once a term is so narrowly defined that it can be true or false it must be agreed upon by two or more people.
In Aristotelean logic most arguments happen because those involved did not agree on their terms, their definitions.
Thus if we agree that Aristotle was the wisest man in history there is little we have to argue about.
There is little point in any discussion between us.
If we agree that Socrates was the wisest man in history then we also have little to disagree about.
There is little point in any discussion between us.
If you believe Aristotle was the wisest man in history and I believe Socrates was the wisest man in history then we disagree about everything.
There is little point in any discussion between us.
Aristotle wanted answers.
Aristotle constructed methods that would force agreements between people who disagreed. If you followed his rules only one truth could exist and the winner of the argument was the one who had demonstrated his was the true argument.
Socrates realized that what people believe to be true is seldom true and that what people believe they know to be real very seldom is. Socrates realized that what is considered knowledge is, like the tree discussed above, a fiction. It does not really exist.
Aristotle represents certainty.
Socrates represents chaos.
Aristotle, like any good preacher, gave the people what they wanted. A feeling of superiority over all lesser beings. These included animals, foreigners, women, and deaf people. He provided simple, easy to master, rules that reinforced this feeling.
Socrates, like any good scientist, sought to find the boundaries of what is known. You cannot explore any concept until you know where the limit of that concept is.
Here we have the basis of logic, both as a workable system of thought, and logic as a failure of reason.
If you believe Aristotle was God’s human gift to Reason, then I am an idiot and there is no point in your paying any attention to anything I have to say.
Your belief is your truth.
If you believe, as I do, that Aristotle did little or nothing to advance humanity and stifled human progress with lousy reasoning, then you must recognize Socrates as a martyr.
Our belief is our truth.
What matters is not that our truth is different.
What matters is that neither of us has the right to force the other to change our truth.
There is no truth.
There is only belief.
And our beliefs are maps.
The maps are not the territories.
Our maps, no matter how useful, are in some way wrong.
If we ask the right questions we may discover where our maps have gone awry.
The law of non-contradiction makes no sense.
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The Map Thinker (tm) 
AJRogersphilosophy has kindly left a link to his blog.
It is a well done blog. If you accept his first three axioms then the rest of his arguments are difficult to dispute.
Worth reading whether you agree with him or not.