Definition:Boolean Interpretation

From ProofWiki
Jump to: navigation, search

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 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 $v: \mathcal L_0 \to \left\{{T, F}\right\}$ inductively, as follows.

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

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

By Boolean Interpretation is Well-Defined, these definitions yield a unique truth value $v \left({\phi}\right)$ 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