Definition:Tautology

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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 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 see



Sources