Law of Identity
The Law of Identify is one of those very simple ideas that is so universal, it's difficult to describe it in words. It is simply the fact that whatever exists, exists in a particular way. Everything that exists has identity.
The identity of an existent is what the existent is. Let's look at an example, say a book. This book in front of me has 535 pages. Each page is made of paper, and are all approximately the same size. The pages have print on them. There's a cover, which is glossy and has a picture of a person. And on and on. What I'm doing with each sentence is identifying the book. Through perception and analysis, I'm able to determine what qualities the book has.
Identity is the fact that allows this process of identification. It simply the fact that the book exists in a particular form, and that means we're able to figure out exactly what form it exists in. So the concept of identity just states simply that a thing is what it is.
Why is this important? Well, fundamentally if it a thing didn't have an identity, it wouldn't be anything at all. If reality were comprised of things that didn't have identity, it would be unknowable. See how basic a concept this is? Everything has identity.
Let's look at some of the non-obvious examples of identity. First, there's your consciousness. I'll explore this concept more, but just think of your own mind. It has identity. It might be incredibly complex, but it exists in a particular way.
Even your ideas or concepts have identity. They don't exist in the material sense, but they have certain qualities. Some of them may be vague and poorly grasped, but those are characteristics too. The same goes for your emotions, or your moods, or even your whims. All of these things are real, and have identity.
The Law of Identity has interesting consequences. Since everything exist in a particular way, there are no contradictions. You can't be both A and not-A at the same time, and in the same respect. If you have a quality, it exists as something, having specific attributes. Either it has the attributes, or it doesn't. So you it can't be both. This is sometimes referred to as "The Law of Excluded Middle".
Earlier I discussed the logic as the "art of non-contradictory identification". This epistemological function is based on a metaphysical premise. That premise is the Law of Identity.
In Objectivists circles, one major application of the Law of Identity is to reject the Copenhagen model of Quantum Physics. You're probably somewhat familiar with it. Light travels as a wave and a particle. You can't measure the velocity and the position at the same time. These, and other ideas, are based on the idea that the light particles do not exist as anything until they are detected. It exists as both a particle and a wave, and yet neither. It has no specific attributes until the probability wave is collapsed. The point of all of this is that it says that something that exists doesn't exist in a particular way. The Law of Identity rejects this.
I'll add quickly that even the scientists who reject the Law of Identity in this case count on it at the same time. They attribute multiple properties to a photon, allowing it to be any number of things at the same time (or none at all, to be precise). But even then, they are attributing qualities to it. They don't say that a light particle can be anything at all. They say it can be several things at once. This attempt to constrain it is an implicit acceptance of the Law of Identity.
Let me give another example. Again later I'll discuss the relationship between our minds and the rest of reality. But one theory is that the world doesn't really exist, and is just a figment of our imaginations. This isn't a complete rejection of reality. The view actually holds that our minds our real, that our imaginations are real, and that they act in a particular way (creating the rest of reality). Once again, they attribute qualities to these things, which relies on the Law of Identity.
The Law of Identity is just another aspect of saying something exists. To say that something exists, or is real, is to say that it exists in a particular form, with particular attributes. You can't have existence without identity, or identity without existence. They're flip sides of the same metaphysical fact. The two concepts are just a change of focus, not a change of topic.