# Definition:Variable

## Definition

A variable is a label which is used to refer to an unspecified object.

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.

## Domain

The collection of all possible objects that a variable may refer to has to be specified.

This collection is the domain of the variable.

### Real Variable

A real variable is a symbol which can stand for any one of a set of real numbers.

### Complex Variable

A complex variable is a symbol which can stand for any one of a set of complex numbers.

## Satisfaction

Let $\map P x$ be a propositional function such that $x$ is a variable with a given domain $S$.

Let a specific element $a$ of $S$ be substituted for $x$ in $\map P x$ such that $\map P a$ is true.

Then $a$ is said to satisfy the propositional function $P$.

## Propositional Logic

A statement variable is a variable which is used to stand for an arbitrary and unspecified statement.

For a statement variable, a lowercase letter is usually used, for example:

$p, q, r, \ldots{}$, and so on

or lowercase Greek letters, for example:

$\phi, \psi, \chi$ and so on.

The citing of a statement variable can be interpreted as an assertion that the statement represented by that symbol is true.

That is:

$p$

means

$p \text { is true}$

## 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.

## Descriptive Statistics

A variable is a characteristic property of all individuals in a population or sample.

It is a categorization of the population such that each individual can be unambiguously described with respect to said variable.

### Quantitative Variable

A quantitative variable is a variable such that:

The variable can be described by numbers
The performing of arithmetical operations on the data is meaningful.

### Qualitative Variable

A qualitative variable is a variable such that:

The variable is not a quantitative variable
The variable describes each individual as either having, or not having, some specific property.

## Value

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

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

## Restricted Variable

A restricted variable is a variable whose values are confined to some only of those of which it is capable.

## Unrestricted Variable

An unrestricted variable is a variable whose values are not confined in any way to some only of those of which it is capable.

## Also known as

When it occurs in a mathematical equation, a variable is often referred to as an unknown.

In the specific context of elementary algebra, the ugly misnomer pronumeral is frequently found in Australia. This was introduced by extension of the concept of a pronoun: a symbol that stands in for a numeral, by which the term number is actually meant.

Thankfully the term appears not to have caught on in general.

## Historical Note

The term variable, as opposed to a constant, was introduced by Gottfried Wilhelm von Leibniz.

However, the concept had already been established by Aristotle in his work on logic.