# Definition:Tautology

## Definition

A tautology is a statement which is always true, independently of any relevant circumstances that could theoretically influence its truth value.

It is epitomised by the statement form:

$p \implies p$

that is:

if $p$ is true then $p$ is true.

An example of a "relevant circumstance" here is the truth value of $p$.

The archetypal tautology is symbolised by $\top$, and referred to as Top.

### Tautologies in Formal Semantics

Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

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

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

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

$\models_{\mathscr M} \phi$

## Also known as

Tautologies are also referred to as logical truths.

## Also defined as

Some sources define a tautology as a statement form which can be epitomised by:

$p \lor \lnot p$

which, while intuitively obvious, it not a universal definition as it does not apply in contexts in which Law of Excluded Middle does not necessarily hold.