Definition:Structure for Predicate Logic/Formal Semantics/Sentence

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Let $\mathcal L_1$ be the language of predicate logic.

The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL}$.

For the purpose of this formal semantics, we consider only sentences instead of all WFFs.

The structures of $\mathrm{PL}$ are said structures for $\mathcal L_1$.

A sentence $\mathbf A$ is declared ($\mathrm{PL}$-)valid in a structure $\mathcal A$ if and only if:

$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) = T$

where $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$ is the value of $\mathbf A$ in $\mathcal A$.

Symbolically, this can be expressed as:

$\mathcal A \models_{\mathrm{PL}} \mathbf A$

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