---
tag: 0B88PQTJ66
kind: definition
name: Universal quantifier
msc: [03B10]
defines: 0GHS47PDAQ
uses: [0T3DPBG981]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_quantification
    locator: lead section
---

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The universal quantification of $P$ over $X$ is the proposition $\forall x \in X, \, P(x)$, read "for all $x$ in $X$, $P(x)$".
It is true when $P(a)$ is true for each $a \in X$, and false otherwise.

## Remarks

When $X$ is empty, $\forall x \in X, \, P(x)$ is true for each $P$. This is a case of [[vacuous truth|0V4QXPGJQK]].
For a finite set $X = \{a_1, \dots, a_n\}$, the statement is equivalent to $P(a_1) \land \dots \land P(a_n)$.

## Explanations

### L1 human

"Every square of a real number is at least $0$" is $\forall x \in \mathbb{R}, \, x^2 \ge 0$.
To prove it, take an arbitrary $x$ and show $x^2 \ge 0$.
To refute it, one [[counterexample|0ED72CTBNP]] is enough.