Definition:Value of Formula under Assignment/Sentence

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Let $\mathcal A$ be a structure for predicate logic.

Let $\mathbf A$ be a sentence in the language of predicate logic.

The value of $\mathbf A$ in $\mathcal A$, denoted $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$, is defined as:

$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) := \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\varnothing}\right]$

where $\varnothing$ is the empty mapping considered as an assignment for $\mathbf A$ and $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\varnothing}\right]$ is the value of $\mathbf A$ under $\varnothing$.

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