Definition:Variable/Predicate Logic

Definition
A variable is a label which is used to refer to an unspecified object. The collection of all possible objects that the variable may refer to has to be specified.

A variable can be identified by means of a symbol, for example, $$x, y, z, A, B ,C, \phi, \psi, \aleph$$. It is often convenient to append a subscript letter or number to distinguish between different objects of a similar type:


 * $$a_0, a_1, a_2, \ldots, a_n; S_\phi, S_{\phi_x}, \ldots$$

The type of symbol used to define a variable is purely conventional. Particular types of object, as they are introduced, frequently have a particular range of symbols specified to define them, but there are no strict rules on the subject.

Predicate Logic
In the context of predicate logic, a variable is often called an object variable or arbitrary name.

As such, it is a symbol which is assigned to an arbitrarily selected object from a given universe of discourse.

The understanding is that (during the scope of the argument to which it is relevant) the arbitrary name could apply equally well to any of the objects in that universe.

Value
A variable $$x$$ may be (temporarily, conceptually) identified with a partcular object.

If so, then that object is called the value of $$x$$.