Definition:Structure for Predicate Logic/Formal Semantics/Sentence

Definition
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 $\mathcal A$ :


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

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

Symbolically, this can be expressed as:


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

Also see

 * Definition:Structure for Predicate Logic
 * Definition:Value of Sentence in Structure for Predicate Logic
 * Definition:Model (Predicate Logic)


 * Definition:Formal Semantics