Universal Quantification

• The notation $\mathrm{\forall }xP(x)$ denotes the universal quantification of $P(x)$
• The symbol $\mathrm{\forall }$ is called the universal quantifier
• We read $\mathrm{\forall }xP(x)$ as ${}^{\prime }forallxP(x{)}^{\prime }$ or ${}^{\prime }foreveryxP(x{)}^{\prime }$
• The domain of x is a set. We have to specify the domain of x
  An element of which $P(x)$ is false is called a counterexample to $\mathrm{\forall }xP(x)$

Note that if the domain is empty, then $\mathrm{\forall }xP(x)$ is true for every propositional function $P(x)$ because there are no elements $x$ in the domain for which $P(x)$ is false.

Counterexamples

When is a statement $\mathrm{\forall }xP(x)$ false?

A statement $\mathrm{\forall }xP(x)$ is false on a given domain, if and only if there is at least one element in $x$ in the domain for which $P(x)$ is false

One way to show that $\mathrm{\forall }xP(x)$ is not true on the given domain is to find a counterexample to the statement $\mathrm{\forall }xP(x)$

A single counterexample is all we need to establish that $\mathrm{\forall }xP(x)$ is false

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 universal quantification $\mathrm{\forall }xP(x)$ expresses the same as the conjunction

because this conjunction is true if and only if all $P(x_1),P(x_2),...,P(x_n)$
are true

• If the domain is empty, the universal quantifier is true