# Definition:Tautology/Formal Semantics

## Definition

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

A logical formula $\phi$ of $\LL$ is a tautology for $\mathscr M$ if and only if:

$\phi$ is valid in every structure $\MM$ of $\mathscr M$

That $\phi$ is a tautology for $\mathscr M$ can be denoted as:

$\models_{\mathscr M} \phi$

## Examples

### Tautology for Boolean Interpretations

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:

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

### Tautology for Predicate Logic

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

Then $\mathbf A$ is a tautology if and only if, for every structure $\AA$ and assignment $\sigma$:

$\AA, \sigma \models_{\mathrm{PL_A} } \mathbf A$

that is, if $\mathbf A$ is valid in every structure $\AA$ and assignment $\sigma$.

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

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

## Also known as

In this context, tautologies are also referred to as (logically) valid formulas.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$, this can easily be confused with a formula that is valid in a single structure, and is therefore discouraged.

## Also denoted as

When the formal semantics under discussion is clear from the context, $\models \phi$ is a common shorthand for $\models_{\mathscr M} \phi$.