Explanation through Atlas Shrugged

Aristole's law of identity has no counterexamples. We can take all class trying to invent some random creative examples that try to prove that "A is NOT A", but it won't happen. I think that a lot of people don't really understand the law and how simple it really is (though one of the most important laws of knowledge). I'm almost finished with the novel Atlas Shurgged (SOOOO GOOD!..by Ayn Rand, creator of the philosophy of Objectivism), and as I was reading it last night, I came across a paragraph that described Aristotle's law perfectly. it helped me to understand-try to object to the law after reading this. The plot's reallyyyyy complicated, but here, the speaker is John Galt, and he's giving a speech to the whole country (which is ready to crash) broadcasted on the radio:

"To exist is to be something, as distinguished from the nothing of non-existence, it is to be an entity of a specific nature made of specific attributes. Centuries ago, the man who was--no matter what his errors--the greatest of your philosophers, has stated the formula defining the concept of existence and the rule of all knowledge: A is A. A thing is itself. You have never grasped the meaning of his statement. I am here to complete it: Existence is Identity, Consciousness is Identification.
"Whatever you choose to consider, be it an object, an attribute, or an action, the law of identity remains the same. A leaf cannot be a stone at the same time, it cannot be all red and all green at the same time, it cannot freeze and burn at the same time. A is A. Or, if you wish it stated in simpler language: You cannot have your cake and eat it, too."