Definition:Structure for Predicate Logic/Formal Semantics
Definition
Let $\mathcal L_1$ be the language of predicate logic.
Formal Semantics for Sentences
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:
- $\mathcal A \models_{\mathrm{PL}} \mathbf A$
Formal Semantics for WFFs
The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL_A}$.
The structures of $\mathrm{PL_A}$ 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 ($\mathrm{PL_A}$-)valid in a structure $\mathcal A$ if and only if:
- $\sigma$ is an assignment for $\mathbf A$
- $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = T$
where $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$ is the value of $\mathbf A$ under $\sigma$.
Symbolically, this can be expressed as one of the following:
- $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$
- $\mathcal A \models_{\mathrm{PL_A}} \mathbf A \left[{\sigma}\right]$
Also see
- Definition:Structure for Predicate Logic
- Definition:Assignment for Structure
- Definition:Model (Predicate Logic)