Equivalent Formula (First-Order Logic)

first-order-logic

Definition

Equivalent Formula (First-Order Logic)

Two formulas $\phi$ and $\psi$ are equivalent, written $\phi \equiv \psi$, if they have the same models:

$$\phi \equiv \psi ⟺((\mathcal{M},\ell )\models \phi \leftrightarrow (\mathcal{M},\ell )\models \psi \text{ for all }\mathcal{M},\ell )$$

Equivalently: $\models \phi \leftrightarrow \psi$. The metametal equivalence ($\equiv$) holds exactly when the object-level biconditional ($\leftrightarrow$) is valid.