Definition:Boolean Interpretation

Definition
Let $\mathcal L_0$ be the language of propositional logic, with vocabulary $\mathcal P_0$.

A boolean interpretation for $\mathcal L_0$ is a propositional function:


 * $v: \mathcal P_0 \to \left\{{T, F}\right\}$

When dealing with explicit situations, it is often convenient to let $v$ be a partial mapping, only defining it on the propositional symbols 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 $v: \mathcal L_0 \to \left\{{T, F}\right\}$ inductively, as follows.

In the following, $f$ denotes the truth function pertaining to its superscript. $P$ and $Q$ denote arbitrary WFFs of $\mathcal L_0$.

By the Principle of Definition by Structural Induction, these definitions yield a unique truth value $v \left({\phi}\right)$ for every WFF $\phi$.

One defines $\phi$ to be valid for $v$ iff $v \left({\phi}\right) = T$, and this is denoted as:


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

Formal Semantics
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 intepretation $v$ iff:


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

Also known as
Some sources simply speak of interpretations.

Other terms in use are valuation and model for propositional logic.

Also see

 * Definition:Language of Propositional Logic
 * Definition:Model (Propositional Logic)