Definition:Tautology/Formal Semantics/Predicate Logic

From ProofWiki
Jump to navigation Jump to search

Definition

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


Sources