Definition:Structure for Predicate Logic/Formal Semantics/Sentence

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Let $\LL_1$ be the language of predicate logic.

The structures for $\LL_1$ can be interpreted as a formal semantics for $\LL_1$, 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 $\LL_1$.

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

$\map {\operatorname{val}_\AA} {\mathbf A} = \T$

where $\map {\operatorname{val}_\AA} {\mathbf A}$ is the value of $\mathbf A$ in $\AA$.

Symbolically, this can be expressed as:

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

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