Universal quantifier

The universal quantification of $P(x)$ for a particular domain, denoted by $\forall xP(x)$, is the proposition "$P(x)$ is true for all values of $x$ from the domain". $\forall$ means "For all".

• The truth value of $\forall xP(x)$ might change if domain changes.
• Domain must always be specified when universal quantifier is used.

For example, take $P(x)$ is "$x>0$":

• For the domain of positive integers, $\forall xP(x)⇝T$.
• For the domain of all integers, $\forall xP(x)⇝F$.

To say “Everyone does not like Jane”, we write that for every $x$, $x$ does not like Jane:

$$(\forall x)(\neg Likes(x,jane))$$

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