Identity

Identity is the fact that a thing is what it is: every existent has a specific nature, a definite set of attributes, and cannot be something else at the same time and in the same respect. It is the third axiom of Objectivist metaphysics, stated in the formula A is A.

“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.”
~ Ayn Rand, "Galt's Speech", Atlas Shrugged

Identity and existence

Identity is not a separate fact added to existence; it is the same fact viewed from another angle. To exist is to exist as something. A thing with no attributes, no nature and no way of being distinguished from anything else would not be an unusual existent but no existent at all.

This is why Rand treated "existence exists" and "A is A" as inseparable, and why the axiom is not the empty tautology it can look like. Its content is negative and substantial: it rules out the possibility that a thing could be and not be, or could act in a way its nature does not permit.

“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, "Galt's Speech", Atlas Shrugged

Causality as identity applied to action

Objectivism's account of causality follows directly. The law of cause and effect is not a separate principle about events succeeding one another; it is the law of identity applied to action. A thing acts according to its nature, and could not act otherwise while remaining what it is.

The unit of causation is therefore the entity, not the event. Things do not simply happen and then cause other happenings; entities act, and what they can do is set by what they are. This is the basis for treating volition as compatible with causality rather than an exception to it.

Its role in logic

Identity is the foundation of logic, which Rand defined as the art of non-contradictory identification. A contradiction is the assertion that something both is and is not what it is, which is why a contradiction cannot exist and why finding one in your thinking means an error in your premises rather than a feature of reality.

“Contradictions do not exist. Whenever you think that you are facing a contradiction, check your premises. You will find that one of them is wrong.”
~ Francisco d'Anconia, in Atlas Shrugged

The same axiom underwrites concept formation. Concepts are possible because existents have definite attributes that can be measured and compared. See Measurement omission and Conceptual Common Denominator.

Common misunderstandings

That "A is A" denies change. It denies that a thing can be and not be in the same respect at the same time. Things change constantly, and change is intelligible precisely because there is something with a nature that undergoes it.
That it is an empty tautology with no consequences. What it excludes is what matters: miracles as violations of a thing's nature, wishing as a means of altering facts, and contradictions as descriptions of reality.
That the "law of identity" is about personal identity. It is a metaphysical principle about all existents. The question of what makes a person the same person over time is a different one.
That accepting it commits you to determinism. Objectivism holds the opposite: since man's nature includes a self-initiated capacity to focus, choice is an instance of a thing acting according to its nature. See Volition.

Discussion

References

Rand, Ayn. "Galt's Speech", Atlas Shrugged. Random House, 1957.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 1.

External links

Identity at the Ayn Rand Lexicon

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