Existential quantifier

z**** Define A quantifier, denoted

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

and read

where \(P\) is the quantified predicate.

(Levin, 2021, sec. 0.2)

Connect

• Universal quantifier

  Equivalent to the universal quantifier under negation by

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

  which is akin De Morgan’s laws, with \(\exists\) generalizing disjunction.

Name

• For some
• There exists
• Some

Explore

There is no greatest integer \(k\) because

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

that is

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