In mathematical logic, a sentence is a formula with no free variables; it follows that, considering model theory, a sentence is either true or false in a given structure.

For example

$(\; \backslash exists\; x)x^2=y$

is not a sentence, because of the presence of the free variable $y$; it will be true in the structure of the real numbers if $y=2$, but not if $y=-2$. On the other hand

$(\backslash forall\; y)(\backslash exists\; x)x^2=y$

is a sentence, and is false in the real numbers.

Mathematical logic

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