# Definition:Tautology/Formal Semantics/Boolean Interpretations

## Definition

Let $\mathbf A$ be a WFF of propositional logic.

Then $\mathbf A$ is called a tautology (for boolean interpretations) if and only if:

$\map v {\mathbf A} = T$

for every boolean interpretation $v$ of $\mathbf A$.

That $\mathbf A$ is a tautology may be denoted as:

$\models_{\mathrm {BI} } \mathbf A$

## Also known as

A tautology in this context may also be described as valid (for boolean interpretations).

On $\mathsf{Pr} \infty \mathsf{fWiki}$, we have chosen to only use validity in the context of a single boolean interpretation.

## Also denoted as

If only boolean interpretations are under discussion, $\models \mathbf A$ is also often encountered.

## Examples

### Example: $\paren {\paren {\paren {\lnot p} \implies q} \implies \paren {\paren {\paren {\lnot p} \implies \paren {\lnot q} } \implies p} }$

The WFF of propositional logic:

$\paren {\paren {\paren {\lnot p} \implies q} \implies \paren {\paren {\paren {\lnot p} \implies \paren {\lnot q} } \implies p} }$

is a tautology.

### Example: $\paren {\paren {\lnot p} \implies \paren {q \lor r} } \iff \paren {\paren {\lnot q} \implies \paren {\paren {\lnot r} \implies p} }$

The WFF of propositional logic:

$\paren {\paren {\lnot p} \implies \paren {q \lor r} } \iff \paren {\paren {\lnot q} \implies \paren {\paren {\lnot r} \implies p} }$

is a tautology.

## Also see

• Results about tautologies can be found here.