Definition:Semantic Equivalence

Definition
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 :


 * $\phi \models_{\mathscr M} \psi$ and $\psi \models_{\mathscr M} \phi$

that is, they are $\mathscr M$-semantic consequences of one another.

Equivalently, $\phi$ and $\psi$ are $\mathscr M$-semantically equivalent, for each $\mathscr M$-structure $\MM$:


 * $\MM \models_{\mathscr M} \phi$ $\MM \models_{\mathscr M} \psi$

Also see

 * Definition:Semantic Consequence


 * Definition:Logical Equivalence
 * Definition:Provable Equivalence