Universal quantifier

Define

\[\begin{equation*}
  \forall x \>\> P(x)
\end{equation*}
\]

and read

“for all/each/every/all/any \(x\), \(P(x)\)”,

where \(P\) is the quantified predicate.

(Levin, 2021, sec. 0.2)

Name

[For] all
[For] each
[For] every

Connect

Existential quantifier

Equivalent to the existential quantifier under negation by

\[\begin{align*}
  \forall x \> \lnot P(x) \> & \equiv \> \lnot \exists x \> P(x) \\
  \lnot \forall x \> P(x) \> & \equiv \> \exists x \> \lnot P(x),
\end{align*}
\]

which is akin De Morgan’s laws, with \(\forall\) generalizing conjunction.

Explore

There is no least integer \(k\) because

\[\begin{equation*}
  \forall k \in \mathbb{Z}
  \quad
  \exists \ell \in \mathbb{Z}
  \quad
  (\ell < k),
\end{equation*}
\]

that is

for every integer \(k\),

there exists some integer \(\ell\)

smaller than \(k\),

such as \(\ell = k - 1\).