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“They proclaim that there is no law of identity, that nothing exists but change, and blank out the fact that change presupposes the concepts of what changes, from what and to what, that without the law of identity no such concept as "change" is possible."”
   ~ Ayn Rand, Atlas Shrugged

A is A

The Axiom of identity is the fact that every thing is itself, instead of anything else. A is not B or C. A is A at all times and in all places.

The Axiom of identity, on its own, is not a particularly useful Concept. It serves instead as the fundamental formula for Logic, induction, deduction and all human knowledge.

In mathematics, each side of the equation must be the same as the other. In Objectivist Epistemology, all statements must be tautological because A is A.

Further Reading

Metaphysics Topics
Axioms: Existence, Consciousness, IdentityCorollary: Causality, Primacy of Existence
Reality: Primacy of Existence, Existents, the Metaphysically Given versus the Man Made