# 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:

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

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 $\LL_0$.

 $\ds \map v \top$ $:=$ $\ds \T$ $\ds \map v \bot$ $:=$ $\ds \F$ $\ds \map v {\neg \phi}$ $:=$ $\ds \map {f^\neg} {\map v \phi}$ $\ds ~=~$ $\ds \begin{cases} \T &: \text{if \map v \phi = \F} \\ \F &: \text{if \map v \phi = \T}\end{cases}$ $\ds \map v {\phi \land \psi}$ $:=$ $\ds \map {f^\land} {\map v \phi, \map v \psi}$ $\ds ~=~$ $\ds \begin{cases} \T &: \text{if \map v \phi = \T = \map v \psi = \T} \\ \F &: \text{otherwise}\end{cases}$ $\ds \map v {\phi \lor \psi}$ $:=$ $\ds \map {f^\lor} {\map v \phi, \map v \psi}$ $\ds ~=~$ $\ds \begin{cases} \F &: \text{if \map v \phi = \F = \map v \psi = \F} \\ \T &: \text{otherwise}\end{cases}$ $\ds \map v {\phi \implies \psi}$ $:=$ $\ds \map {f^\Rightarrow} {\map v \phi, \map v \psi}$ $\ds ~=~$ $\ds \begin{cases} \T &: \text{if \map v \phi = \F or \map v \psi = \T} \\ \F &: \text{otherwise}\end{cases}$ $\ds \map v {\phi \iff \psi}$ $:=$ $\ds \map {f^\Leftrightarrow} {\map v \phi, \map v \psi}$ $\ds ~=~$ $\ds \begin{cases} \T &: \text{if \map v \phi = \map v \psi} \\ \F &: \text{otherwise}\end{cases}$

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

### Boolean Interpretation for Formula

Let $v: \LL_0 \to \set {\T, \F}$ be a (partial) boolean interpretation.

Let $\phi$ be a WFF of propositional logic.

Then $v$ is called a boolean interpretation for $\phi$ if and only if $v$ is defined at $\phi$.

Otherwise, $v$ is called a partial (boolean) interpretation for $\phi$.

### Boolean Interpretation for Set of Formulas

Let $v: \LL_0 \to \set {\T, \F}$ be a (partial) boolean interpretation.

Let $\FF$ be a set of WFFs of $\LL_0$.

Then $v$ is called a boolean interpretation for $\FF$ if and only if $v$ is defined on $\FF$.

Otherwise, $v$ is called a partial (boolean) interpretation for $\FF$.

### Truth Value

Let $\phi$ be a WFF of propositional logic.

Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation of $\phi$.

The truth value of $\phi$ under $v$ is $v (\phi)$.

### Formal Semantics

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:

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

## Also known as

Some sources simply speak of interpretations.

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

## Also see

• Results about boolean interpretations can be found here.

## Source of Name

This entry was named for George Boole.