After learning logical laws for quantified statements, we now study one of the most important transformations in predicate logic: negation. Negation of quantified statements changes the quantifier and negates the predicate. This is the formal version of the ordinary idea that “not all” means “at least one not,” and “none” means “all not.” In this lesson, negation of quantified statements will be simplified using quantifier rules, De Morgan’s laws, and the negation of implication.

For any predicate $p(x)$, $$\begin{equation*} \neg[\forall x\,p(x)]\leftrightarrow\exists x\,\neg p(x). \end{equation*}$$ This means that “not all elements satisfy $p$” is equivalent to “there exists at least one element that does not satisfy $p$.”