Definition:Structure for Predicate Logic/Formal Semantics
Definition
Let $\LL_1$ be the language of predicate logic.
Formal Semantics for Sentences
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:
- $\AA \models_{\mathrm{PL} } \mathbf A$
Formal Semantics for WFFs
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 $\AA$ if and only if:
- $\sigma$ 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 Structure
- Definition:Model (Predicate Logic)