A predicate is a statement with a hole: it becomes true or false only once the hole is filled, or quantified.

The predicate

Definition

A proposition is a statement that is true or false: "7 is prime" is true, "8 is prime" is false.

A predicate has no such property. " $x$ is prime" is neither true nor false: it depends on $x$ . It is a statement with a hole, which returns a truth value once the hole is filled.

Free variable, bound variable

As long as the variable stays free, the statement has no truth value. Two operations close it:

Operation	Example	Result
Instantiate	$P(7)$	True
Quantify universally	$\forall x, P(x)$	False over the integers
Quantify existentially	$\exists x, P(x)$	True over the integers

A quantified variable is said to be bound. A statement with no free variable is a proposition — it has a truth value, and only one.

The order of quantifiers

Two quantifiers of different kinds do not commute. Over the integers, with $S(x, y)$ for " $y$ is greater than $x$ ":

$\forall x\, \exists y,\ S(x, y) \quad \text{is true}$

$\exists y\, \forall x,\ S(x, y) \quad \text{is false}$

The first says: for each integer there exists a greater one — and $y$ may depend on $x$ . The second says: there exists an integer greater than all — a largest integer, which does not exist.

The same symbols, in reverse order, state a truth and a falsehood. It is the same mechanism that makes the definition of a limit non-commutative.

Negating a quantified statement

Negation crosses quantifiers by swapping them:

$\neg\,\forall x,\ P(x) \iff \exists x,\ \neg P(x)$

$\neg\,\exists x,\ P(x) \iff \forall x,\ \neg P(x)$

Negating "all are prime" does not give "none is prime", but "there exists one that is not". A single counterexample refutes a universal; refuting an existential requires sweeping the whole domain.

The domain is part of the statement

A predicate can only be judged over a declared domain. " $\exists x,\ x^2 = 2$ " is false over $\mathbb{Q}$ and true over $\mathbb{R}$ : the formula has not changed, the universe has.

To omit the domain is to leave the statement undetermined — not ambiguous in meaning, but devoid of any truth value.

Summary

A predicate becomes a proposition when its variables are fixed or quantified over a declared domain.