Definition:Structure for Predicate Logic/Formal Semantics/Sentence
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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 $\mathrm{PL}$.
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:
- $\AA \models_{\mathrm{PL} } \mathbf A$
Also see
- Definition:Structure for Predicate Logic
- Definition:Value of Sentence in Structure for Predicate Logic
- Definition:Model (Predicate Logic)
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.7$ First-Order Logic Semantics: Definition $\text{II.7.8}$