# Definition:Boolean Interpretation

## Contents

## 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 $\LL_0$.

\(\displaystyle \map v \top\) | \(:=\) | \(\displaystyle \T\) | |||||||||||

\(\displaystyle \map v \bot\) | \(:=\) | \(\displaystyle \F\) | |||||||||||

\(\displaystyle \map v {\neg \phi}\) | \(:=\) | \(\displaystyle \map {f^\neg} {\map v \phi}\) | \(\displaystyle ~=~\) | \(\displaystyle \begin{cases} \T &: \text{if $\map v \phi = \F$} \\ \F &: \text{if $\map v \phi = \T$}\end{cases}\) | |||||||||

\(\displaystyle \map v {\phi \land \psi}\) | \(:=\) | \(\displaystyle \map {f^\land} {\map v \phi, \map v \psi}\) | \(\displaystyle ~=~\) | \(\displaystyle \begin{cases} \T &: \text{if $\map v \phi = \T = \map v \psi = \T$} \\ \F &: \text{otherwise}\end{cases}\) | |||||||||

\(\displaystyle \map v {\phi \lor \psi}\) | \(:=\) | \(\displaystyle \map {f^\lor} {\map v \phi, \map v \psi}\) | \(\displaystyle ~=~\) | \(\displaystyle \begin{cases} \F &: \text{if $\map v \phi = \F = \map v \psi = \F$} \\ \T &: \text{otherwise}\end{cases}\) | |||||||||

\(\displaystyle \map v {\phi \implies \psi}\) | \(:=\) | \(\displaystyle \map {f^\Rightarrow} {\map v \phi, \map v \psi}\) | \(\displaystyle ~=~\) | \(\displaystyle \begin{cases} \T &: \text{if $\map v \phi = \F$ or $\map v \psi = \T$} \\ \F &: \text{otherwise}\end{cases}\) | |||||||||

\(\displaystyle \map v {\phi \iff \psi}\) | \(:=\) | \(\displaystyle \map {f^\Leftrightarrow} {\map v \phi, \map v \psi}\) | \(\displaystyle ~=~\) | \(\displaystyle \begin{cases} \T &: \text{if $\map v \phi = \map v \psi$} \\ \F &: \text{otherwise}\end{cases}\) |

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 $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a (partial) boolean interpretation.

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

Then $v$ is called a **boolean interpretation for $\phi$** iff $v$ is defined at $\phi$.

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

### Boolean Interpretation for Set of Formulas

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

Let $\mathcal F$ be a set of WFFs of $\mathcal L_0$.

Then $v$ is called a **boolean interpretation for $\mathcal F$** iff $v$ is defined on $\mathcal F$.

Otherwise, $v$ is called a **partial (boolean) interpretation for $\mathcal F$**.

### 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 $\mathcal L_0$ 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:

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

## 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.

## Sources

- 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.3$: Boolean interpretations: Definition $2.3.1$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.5$: Semantics of Propositional Logic - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.5$: Semantics of Propositional Logic - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.4.1$: The meaning of logical connectives