Definition:Tautology/Formal Semantics

Definition
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 $\mathscr M$ iff:


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

However, on, 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 $\models_{\mathscr M} \phi$.

Also see

 * Definition:Top (Logic), a symbol often used to represent tautologies in logical languages.
 * Definition:Contradiction
 * Definition:Contingent Statement


 * Definition:Validity