Definition:Boolean Interpretation/Formal Semantics

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:

$v \not\models_{\mathrm{BI}} \phi$