Definition:Boolean Interpretation/Formal Semantics
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Definition
Let $\LL_0$ be the language of propositional logic.
The boolean interpretations for $\LL_0$ can be interpreted as a formal semantics for $\LL_0$, which we denote by $\mathrm{BI}$.
The structures of $\mathrm{BI}$ are the boolean interpretations.
A WFF $\phi$ is declared ($\mathrm{BI}$-)valid in a boolean interpretation $v$ if and only if:
- $\map v \phi = \T$
Symbolically, this can be expressed as:
- $v \models_{\mathrm{BI}} \phi$
Invalid
$\phi$ is declared ($\mathrm{BI}$-)invalid in a boolean interpretation $v$ if and only if:
- $\map v \phi = \F$
Symbolically, this can be expressed as:
- $v \not\models_{\mathrm{BI}} \phi$
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