Identities

Why

We can give the same object two different names.

Definition

An object is itself. If the object denoted by one name is the same as the object denoted by a second name, then we say that the two names are equal. The object associated with a name is the identity of the name.

Let $A$ denote an object and let $B$ denote an object. Here we are using $A$ and $B$ as placeholders. They are names for objects, but we do not know—or care—which objects. We say “$A$ equals $B$” as a shorthand for “the object denoted by $A$ is the same as the object denoted by $B$.” In other words, $A$ and $B$ are two names for the same object.

Symmetry

Let $A$ denote an object and let $B$ denote an object. “$A$ equals $B$” means the same as “$B$ equals $A$”. The identity of the names is not dependent on the order in which the names are given. We call this the symmetry of identity. It means we can switch the spots of $A$ and $B$ and say the same thing. In other words, there are two ways to make the statement.

Reflexivity

Let $A$ denote an object. Since every object is the same as itself, the object denoted by $A$ is the same as the object denoted by $A$. We say “$A$ equals $A$”. In other words, every name equals itself. This fact is called the reflexivity of identity. A name is equal to itself because an object is itself.