Definition:Tautology/Formal Semantics/Predicate Logic

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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 $\models_{\mathrm{PL_A} } \phi$.