Existential quantifier

The existential quantification of $P(x)$ for a particular domain, denoted by $\exists xP(x)$, is the proposition that "there exists a value for $x$ in the domain such that $P(x)$ is true".

• The truth value of $\exists xP(x)$ might change when domain changes.
• Domain must always be specified when used.

For example, take $P(x)$ is "$x<-3$":

• If the domain is all integers, then it is true.
  • We simply need to give one example, such as $x=-5$ then $P(-5)$ is true.
• If the domain is the positive integers, then it is false.

To say “(at least one) car is yellow”, we write that for some $x$, $x$ is yellow:

$$(\exists x)(Colour(x,yellow))$$

In general, the notation for existential quantifiers is $(\exists x)(\phi)$, where $\phi$ is a formula in which $x$ and possible other variables may appear..