Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula

Definition
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 $\mathrm{PL_A}$.

The structures of $\mathrm{PL_A}$ are pairs $\struct {\AA, \sigma}$, where:


 * $\AA$ is a structure for $\LL_1$
 * $\sigma$ is an assignment for $\AA$

A WFF $\mathbf A$ is declared ($\mathrm{PL_A}$-)valid in a structure $\AA$ :


 * $\sigma$ is an assignment for $\mathbf A$
 * $\map {\operatorname{val}_\AA} {\mathbf A} \sqbrk \sigma = \T$

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

Symbolically, this can be expressed as one of the following:


 * $\AA, \sigma \models_{\mathrm{PL_A} } \mathbf A$


 * $\AA \models_{\mathrm{PL_A} } \mathbf A \sqbrk \sigma$

Also see

 * Definition:Structure for Predicate Logic
 * Definition:Assignment for Formula
 * Definition:Model (Predicate Logic)


 * Definition:Formal Semantics