\begin{definition}[Definition:Boolean Interpretation]
Let $\LL_0$ be the language of propositional logic, with vocabulary $\PP_0$.
A '''boolean interpretation''' for $\LL_0$ is a propositional function:
:$v: \PP_0 \to \set {\T, \F}$
When dealing with explicit situations, it is often convenient to let $v$ be a partial mapping, only defining it on the part of the vocabulary one is currently dealing with.
In such cases $v$ may be called a '''partial boolean interpretation'''; note that this term is taken to include bona fide '''boolean interpretations''' as well.
Next, one extends the '''boolean interpretation''' $v$ to a (partial) mapping $v: \LL_0 \to \set {\T, \F}$ inductively, as follows.
In the following, $f$ denotes the truth function pertaining to its superscript, while $\phi$ and $\psi$ denote arbitrary WFFs of $\LL_0$.
{{begin-eqn}}
{{eqn | l = \map v \top
      | o = :=
      | r = \T
}}
{{eqn | l = \map v \bot
      | o = :=
      | r = \F
}}
{{eqn | l = \map v {\neg \phi}
      | o = :=
      | r = \map {f^\neg} {\map v \phi}
      | rr= ~=~
      | rrr=\begin{cases} \T &: \text{if $\map v \phi = \F$} \\ \F &: \text{if $\map v \phi = \T$}\end{cases}
}}
{{eqn | l = \map v {\phi \land \psi} 
      | o = :=
      | r = \map {f^\land} {\map v \phi, \map v \psi}
      | rr= ~=~
      | rrr=\begin{cases} \T &: \text{if $\map v \phi = \T = \map v \psi = \T$} \\ \F &: \text{otherwise}\end{cases}
}}
{{eqn | l = \map v {\phi \lor \psi}
      | o = :=
      | r = \map {f^\lor} {\map v \phi, \map v \psi}
      | rr= ~=~
      | rrr=\begin{cases} \F &: \text{if $\map v \phi = \F = \map v \psi = \F$} \\ \T &: \text{otherwise}\end{cases}
}}
{{eqn | l = \map v {\phi \implies \psi} 
      | o = :=
      | r = \map {f^\Rightarrow} {\map v \phi, \map v \psi}
      | rr= ~=~
      | rrr=\begin{cases} \T &: \text{if $\map v \phi = \F$ or $\map v \psi = \T$} \\ \F &: \text{otherwise}\end{cases}
}}
{{eqn | l = \map v {\phi \iff \psi} 
      | o = :=
      | r = \map {f^\Leftrightarrow} {\map v \phi, \map v \psi}
      | rr= ~=~
      | rrr=\begin{cases} \T &: \text{if $\map v \phi = \map v \psi$} \\ \F &: \text{otherwise}\end{cases}
}}
{{end-eqn}}
By Boolean Interpretation is Well-Defined, these definitions yield a unique truth value $\map v \phi$ for every WFF $\phi$.
\end{definition}