Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| formula logic |
✓ |
$\exists y \, (y > x)$ is a formula, because $\exists y \, (y > x)$ applies the quantifier $\exists y$ to the formula $y > x$.
|
| sentence |
✗ |
$\exists y \, (y > x)$ is not a sentence, although $\exists y \, (y > x)$ has a quantifier: the quantifier binds $y$, but $x$ is free, so the truth of $\exists y \, (y > x)$ depends on the value of $x$ (for example, if $x$ and $y$ range over the natural numbers, then the formula is true for every $x$, and if $x$ and $y$ range over $\setdef{1, 2, 3}$, then the formula is false for $x = 3$).
|
Distinctions using this example
- Formula vs. sentence: $\exists y \, (y > x)$ shows that a formula with a quantifier can still fail to be a sentence: the quantifier binds $y$ but leaves $x$ free.
All examples · All distinctions · All definitions