Aristotle's Law of Identity

The granddaddy of 'em all. Aristotle's Law Of Identity, or at least what got called by that name in the Aristotelean/Scholastic tradition, is usually rendered as:

 A is A.

but the intended meaning is that everything possesses a certain, definite nature. Everything is a certain way.

Oh, you want to know why "A is A" seems to some people like a reasonable way to express this thought. Okay, I'll tell you, but you'll have to learn something about Aristotle's theory of proposition.

The first A is the subject of the proposition. You can fill in any thing you like here: your cat Purrcollator, your first-grade teacher's left pinky fingernail, the rebelliousness of youth, the organization that decides where to hold the next Olympic games, etc.

The second A is the predicate. A predicate is a certain way that applies or doesn't apply to the subject. For example, the sort of curved shape and irregular ridges of the aforementioned pinky fingernail, together with its translucence, its chemical structure, its hardness, and everything else about it.

A Categorical Proposition either affirms or denies that a certain predicate applies to a certain subject. "A is A" affirms that, no matter what thing you are talking about, it has a certain nature--its own certain nature, its own way that it is. This thing A has A-nature.

The main idea, though, is the notion of a "certain way". The Law Of Identity is that everything has one. Or in other words, Big Omega is right out.

In Aristotelean philosophy, the Law Of Identity is held to identify an all-pervasive principle of nature, which we try to reflect in rational thought. The things have natures, and our job as thinkers is to comprehend those natures. The very nature of affirming and denying--propositions with subjects and predicates--reflects the existence of things and their identities. Without understanding the Law Of Identity in some form, it would be very difficult to understand anything about reason--that is, anything about propositions, terms, inference, classification, definition, and so on. Denying the Law Of Identity would implicitly deny the validity of all rational thought. It wouldn't be about anything.

Sameness

Since the word identity sounds like identical, deriving, as it does, from the Latin idem meaning "same", people sometimes guess that the Law Of Identity is:

 Everything is the same as itself.

or:

 Everything is identical to itself.

or even better:

 Everything is interchangeable with itself.

Indeed everything is, but that's a rather silly law, hardly material for a foundation of rational thought.

Symbolic-Logic Misreadings

The "interchangeableness" reading seems to derive from a theory of logic that says that thinking is purely a matter of substituting symbol-strings for other symbol-strings according to mechanical rules. In Aristotelean philosophy, thinking is getting your mind in tune with the nature of things, and logic is the study of how that process works and how to do it well.

Another definition of the Law Of Identity that you sometimes see is:

 P -> P

that is, from any proposition P, you can derive P. P implies P, for any proposition P. Of course the Aristotelean Law Of Identity is not a law about deriving propositions from other propositions, it's a law about things.

Mathematical Laws of Identity

Completely unrelated are mathematical laws that postulate a binary operation O and a constant I, such that x O I = x for all x.

For example, the multiplicative law of identity says that any number multiplied by 1 equals that same number.