After translating quantifiers in mathematics and programming, we now study their exact truth conditions. Truth conditions for quantified statements tell us how to prove or disprove existential and universal statements. An existential statement needs one witness, while a universal statement must survive every element of the universe. In this lesson, truth conditions for quantified statements will be used to translate, evaluate, and find counterexamples.

Let $p(x)$ be an open statement over a universe $U$. The statement $\exists x\,p(x)$ is true if $p(a)$ is true for at least one $a\in U$. It is false if $p(a)$ is false for every $a\in U$.