Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| proposition |
True if $x$ ranges over $\reals$; false if $x$ ranges over the rational numbers
|
$\exists x \, (x^2 = 2)$ expresses a proposition only once the set that $x$ ranges over is chosen: over $\reals$, the number $\sqrt{2}$ satisfies $x^2 = 2$, and over the rational numbers, no number does.
|
| sentence |
✓ |
$\exists x \, (x^2 = 2)$ is a sentence, because the quantifier $\exists x$ binds the only variable $x$.
|
| formula logic |
✓ |
$\exists x \, (x^2 = 2)$ is a formula, because $\exists x \, (x^2 = 2)$ applies the quantifier $\exists x$ to the formula $x^2 = 2$.
|
Distinctions using this example
- Sentence vs. proposition: The sentence $\exists x \, (x^2 = 2)$ expresses a true proposition when $x$ ranges over the real numbers and a false proposition when $x$ ranges over the rational numbers.
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