\begin{definition}[Definition:Semantic Equivalence]
Let $\mathscr M$ be a formal semantics for a formal language $\LL$.
Let $\phi, \psi$ be $\LL$-WFFs.
Then $\phi$ and $\psi$ are $\mathscr M$-'''semantically equivalent''' {{iff}}:
:$\phi \models_{\mathscr M} \psi$ and $\psi \models_{\mathscr M} \phi$
that is, {{iff}} they are $\mathscr M$-semantic consequences of one another.
Equivalently, $\phi$ and $\psi$ are $\mathscr M$-'''semantically equivalent''' {{iff}}, for each $\mathscr M$-structure $\MM$:
:$\MM \models_{\mathscr M} \phi$ {{iff}} $\MM \models_{\mathscr M} \psi$
{{transclude:Definition:Semantic Equivalence/Boolean Interpretations
  |section = tc
  |title = Semantic Equivalence for Boolean Interpretations
  |header = 3
  |link = true
  |increase = 1
  |foldable = true
}}
{{transclude:Definition:Semantic Equivalence/Predicate Logic
  |section = tc
  |title = Semantic Equivalence for Predicate Logic
  |header = 3
  |link = true
  |increase = 1
  |foldable = true
}}
\end{definition}