Definition:Boolean Interpretation

Definition
Let $\LL_0$ be the language of propositional logic, with vocabulary $\PP_0$.

A boolean interpretation for $\LL_0$ is a propositional function:


 * $v: \PP_0 \to \set {\text T, \text F}$

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 $v: \LL_0 \to \set {\mathrm T, \mathrm F}$ inductively, as follows.

In the following, $f$ denotes the truth function pertaining to its superscript, while $\phi$ and $\psi$ denote arbitrary WFFs of $\LL_0$.

By Boolean Interpretation is Well-Defined, these definitions yield a unique truth value $\map v \phi$ for every WFF $\phi$.

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 (Boolean Interpretations)