Definition:Tautology/Formal Semantics/Predicate Logic

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

Then $\mathbf A$ is a tautology, 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, 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 $\models_{\mathrm{PL_A} } \phi$.