Law of Non-Contradiction

A ? ¬A

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The Law of Non-Contradiction, as stated by Aristotle: 

"One cannot say of something that it is and that it is not in the same respect and at the same time."

Often, this is expressed in the formula: A ? ¬A, where "¬A" signifies "not A" or "not having quality A". (To prevent a common error, understand that it does NOT mean "the subset of everything except A." For example, to say that "an animal that is a cat cannot be, at the same time, a dog" is NOT an application of the law of non-contradiction. To say that "an animal that is a cat cannot, at the same time, also be a 'not cat'" IS to apply the law of non-contradiction.)

Worded a bit more clearly:

Nothing can both have the quality of A and lack the quality of A at the same time.

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This video does an excellent job of tackling the philosophical mess of the ontological argument and clarifying the nature of logic and mathematics incorporating A ? ¬A to help illustrate the point.:

 

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I originally found the text above here... It was the first part of this article (emphasis and formatting added by me):
(the vid I found later)

Morality Is Relative But Not Subjective

http://bigthink.com/ideas/41972

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