Existential Quantification

The $\text{existential quantification}$ of $P(x)$ is the statement

• The notation $\mathrm{\exists }xP(x)$ denotes the existential quantification of $P(x)$
• The symbol $\mathrm{\exists }$ is called the $\text{existential quantifier}$
• We read $\mathrm{\exists }xP(x)$ as $\text{‘}\text{there exists an }xP(x)\text{’}$ or $\text{‘}\text{there is an }xP(x)\text{’}$

The existential quantification $\mathrm{\exists }xP(x)$ can be expressed as

• 'There is an $x$ such that $P(x)$'
• 'There is at least one $x$ such that $P(x)$'
• 'For some $x$ $P(x)$'

Remarks

• A domain must always be specified when a statement $\mathrm{\exists }xP(x)$ is used
• The meaning of $\mathrm{\exists }xP(x)$ changes when the domain changes
• Without specifying the domain, the statement $\mathrm{\exists }xP(x)$ has no meaning

Finite Domains

When the domain is $finite$, i.e. when all the elements in the domain can be listed, say, $x_1,x_2,...,x_n$ it follows that the existential quantification $\mathrm{\exists }xP(x)$ expresses the same as the disjunction

because this disjunction is true if and only if at least one $P(x_1),P(x_2),...,P(x_n)$ is true

• If the domain is empty, the existential quantifier is false