From The New York Times, April 9, 1961. Excerpt:
Since his baptism in medieval times, Aristotle has served many strange purposes. None have been odder than this sacramental alliance, so to speak, of Aristotle with Adam Smith. The extraordinary virtues Miss Rand finds in the law that A is A suggests that she is unaware that logical principles by themselves can test only consistency. They cannot establish truth . . . . Swearing fidelity to Aristotle, Miss Rand claims to deduce not only matters of fact from logic but, with as little warrant, ethical rules and economic truths as well. As she understands them, the laws of logic license her in proclaiming that “existence exists,” which is very much like saying that the law of gravitation is heavy and the formula of sugar sweet.
One of the few things that almost all professional philosophers agree on is that Ayn Rand makes mischief with the Law of Identity.
But the estimable Professor Hook makes a mistake above. It is of course true that the law of gravitation is not itself subject to the law of gravitation: it is not heavy or the opposite. This comparison would be apt, however, only if Rand thought that existence is something distinct from existents. But when she says that existence exists, she does not mean that there is something called 'existence' which is distinct from existing things and that it too exists. She is using 'existence' as a term that refers to existents collectively, similarly as when we use 'humanity' to refer collectively to human beings, as opposed to using it to refer to the being-human of human beings.
When Rand says that existence exists, what she means is that each existing thing exists and has the nature it has independently of any consciousness, including divine consciousness. She is thus an extreme metaphysical realist.
Unfortunately, Rand tries to squeeze this extreme thesis from the logical truth, A = A. And so Hook and almost all professional philosophers are right to critize her for her metaphysical chutzpah.
I lay this all out in painful detail in Ayn Rand on "Existence Exists."