Variable Binding

Definition

Variable binding occurs when a quantifier controls a variable inside its scope.

Example:

For every x, P(x)

The quantifier:

For every x

binds the variable x in:

P(x)

A bound variable is controlled by a quantifier.

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Bound Variable

In:

For every x, Human(x)

the variable x is bound.

It does not refer to one specific object.

It ranges over the domain.

The statement says:

every object in the domain is human

A bound variable ranges over the domain specified by the quantifier.

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Free Variable

A free variable is not controlled by a quantifier.

Example:

Human(x)

Here x is free.

The expression is not yet a complete proposition unless x is assigned.

Compare:

For every x, Human(x)

Now x is bound.

Free variables need assignment or binding.

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Same Symbol, Different Binding

The same variable symbol can be reused in separate quantified expressions.

Example:

For every x, P(x)
There exists x such that Q(x)

The two uses of x are controlled by different quantifiers.

They do not automatically refer to the same object.

Binding is local to the quantifier's scope.

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Why Binding Matters

Variable binding prevents confusion about what a variable refers to.

Example:

For every x, P(x) -> Q(x)

We need to know whether both P(x) and Q(x) are inside the scope of the same x.

If binding is unclear, the meaning is unclear.

Binding connects variables to the quantifiers that control them.

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Final Summary

• Variable binding happens when a quantifier controls a variable.
• A bound variable is inside the quantifier's scope.
• A free variable is not controlled by a quantifier.
• Free variables need assignment or binding.
• The same symbol can be reused under different quantifiers.
• Binding clarifies what variables refer to in logical expressions.