Definition:Structure for Predicate Logic/Formal Semantics

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 $\mathrm{PL}$.

The structures of $\mathrm{PL}$ are pairs $\left({\mathcal A, \sigma}\right)$, where:


 * $\mathcal A$ is a structure for $\mathcal L_1$
 * $\sigma$ is an assignment for $\mathcal A$

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


 * $\sigma$ is an assignment for $\mathbf A$
 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = T$

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

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


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


 * $\mathcal A \models_{\mathrm{PL}} \mathbf A \left[{\sigma}\right]$

When $\sigma$ is the empty mapping (and consequently $\mathbf A$ is a sentence), it is dropped from the notation:


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

Also see

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


 * Definition:Formal Semantics