A mathematical statement is a claim that is either true or false.

Examples include

$$2+3=5$$

and

$$x^2\ge 0\text{ for every real }x.$$

An expression containing an unspecified variable, such as $x>3$, becomes a definite statement only after the variable is fixed or quantified.

Universal and existential quantifiers

The universal quantifier means “for every”, while the existential quantifier means “there exists”. If $S$ denotes the allowed domain of values and $x\in S$ means that $x$ belongs to that domain, then

$$\forall x\in S,\ P(x)$$

means that $P(x)$ is true for every allowed $x$, while

$$\exists x\in S\text{ such that }P(x)$$

means that at least one allowed $x$ makes $P(x)$ true.

The order of quantifiers matters. For example,

$$\forall x,\exists y;(y>x)$$

is very different from

$$\exists y,\forall x;(y>x).$$

Negating quantified statements

Negation swaps the quantifier and negates the property:

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

$$\neg(\exists x,P(x))\equiv\forall x,\neg P(x).$$

Quantifiers make explicit what a theorem claims and what evidence would refute it.