# Definition:Boolean Interpretation/Formal Semantics

## Definition

Let $\mathcal L_0$ be the language of propositional logic.

The boolean interpretations for $\mathcal L_0$ can be interpreted as a formal semantics for $\mathcal L_0$, 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 $v$ iff:

$v \left({\phi}\right) = T$

Symbolically, this can be expressed as:

$v \models_{\mathrm{BI}} \phi$

### Invalid

$\phi$ is declared ($\mathrm{BI}$-)invalid in a boolean interpretation $v$ iff:

$v \left({\phi}\right) = F$

Symbolically, this can be expressed as:

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