= Semantic consequence under an assignment
{title2=$\Gamma\models\varphi[v]$}
In a fixed <first-order structure>, the satisfaction implication at a <first-order variable assignment> $v$ says that if all formulas of $\Gamma$ hold at $v$, then $\varphi$ holds at $v$. Semantic consequence requires that implication in every relevant structure and assignment. The assignment's codomain belongs to its structure; different domains are not silently assigned one common map. When all formulas are <first-order sentences>, assignments do not affect any of their truth values.
Back to article page
