Universal Sentence — Definition, Formula & Examples

A universal sentence is a logical statement that claims something is true for every element in a given domain. It uses the universal quantifier

\forall

(read "for all") to express that a property or condition applies without exception.

A universal sentence is a closed formula of the form $\forall x\, P(x)$ , where $P(x)$ is a predicate and the quantifier $\forall$ binds the variable $x$ over a specified domain of discourse $D$ . The sentence is true if and only if $P(a)$ holds for every element $a \in D$ .

Key Formula

\forall x \in D,\; P(x)

Where:

$\forall$ = Universal quantifier, read as "for all" or "for every"
$x$ = A variable ranging over the domain
$D$ = The domain of discourse (the set of objects under consideration)
$P(x)$ = A predicate — a statement whose truth depends on x

How It Works

To evaluate a universal sentence, you check whether the predicate is satisfied by every object in the domain. If even one object fails to satisfy it, the sentence is false — that single failure is a counterexample. In practice, universal sentences often take the form of conditionals:

\forall x\,(Q(x) \rightarrow R(x))

, meaning "for all

x

, if

Q(x)

then

R(x)

." Proving a universal sentence typically requires a general argument (such as a direct proof or proof by contradiction) that covers all cases, whereas disproving it requires only one counterexample.

Example

Problem: Let the domain be all integers. Determine the truth value of the universal sentence: "For every integer

n

n^2 \geq 0

Formalize: Write the sentence in symbolic form with the universal quantifier.

\forall n \in \mathbb{Z},\; n^2 \geq 0

Check cases: Consider representative values: if

n = 3

, then

n^2 = 9 \geq 0

. If

n = 0

, then

n^2 = 0 \geq 0

. If

n = -5

, then

n^2 = 25 \geq 0

. Squaring any integer always yields a non-negative result.

Conclude: Because no integer can produce a negative square, the predicate holds for every element of the domain.

Answer: The universal sentence

\forall n \in \mathbb{Z},\; n^2 \geq 0

is true.

Why It Matters

Universal sentences are the backbone of mathematical theorems — nearly every theorem asserts that something holds for all objects meeting certain conditions. Learning to read, write, and negate them correctly is essential in proof-based courses like discrete mathematics, real analysis, and abstract algebra.

Common Mistakes

Mistake: Believing that checking a few examples proves a universal sentence is true.

Correction: Examples can only disprove a universal sentence (via counterexample). To prove one, you need a general argument that covers every element in the domain.

Related Terms

Counterexample — One counterexample disproves a universal sentence
Conditional — Universal sentences often contain conditionals
Conjecture — Unproven universal claim awaiting proof
Axiom — Axioms are accepted universal truths
Contrapositive — Used to prove universal conditional sentences
Conclusion — The result derived within a universal statement