Universal rules handle statements about every element. Existential rules handle statements about at least one element. Quantifier laws and existential rules must be used carefully, because an existentially chosen element is a particular witness, not an arbitrary object. In this lesson, existential specification, existential generalization, and important quantifier laws are proved and applied to quantified arguments.

The Rule of Existential Specification states that if $\exists x\,p(x)$ is true, then there is at least one element of the universe for which $p(x)$ is true. We may name such an element $a$ and write $p(a)$, provided that $a$ is treated as a particular element whose existence is guaranteed, not as an arbitrary element.