Definition:Semantic Consequence/Predicate Logic

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Definition

Let $\FF$ be a collection of WFFs of predicate logic.

Then a WFF $\mathbf A$ is a semantic consequence of $\FF$ if and only if:

$\AA \models_{\mathrm{PL} } \FF$ implies $\AA \models_{\mathrm{PL} } \mathbf A$

for all structures $\AA$, where $\models_{\mathrm{PL} }$ is the models relation.

Notation

That $\mathbf A$ is a semantic consequence of $\FF$ is denoted as:

$\FF \models_{\mathrm{PL} } \mathbf A$