Logical Equivalence

Logical equivalence means that two formulae have the same meaning (or semantics). That is, F is true whenever G is true, and vice versa. More formally, logical equivalence two formulae logically entail each other.

Definition §

Logical Equivalence

Let $F$ and $G$ be two formula. Then:

$$(F\equiv G)\text{ iff }[(F\models G)\text{ and }(G\models F)]$$

In English: F and G are logically equivalent if, and only if, F logically entails G and vice versa.

• $\equiv$: Logical equivalence
• $F\models G$: $F$ entails $G$ or $G$ is a semantic consequence of $F$

Theorems §

T1: In propositional logic, bi-implication under every model is equivalent to logical equivalence

Let $F$ and $G$ be two propositions (which are a type of formulae). Then:

$$(F\equiv G)\text{ iff }(\models (F\leftrightarrow G))$$

In English: F and G are logically equivalent if, and only if, under every model, F and G are materially equivalent (or they imply each other). Note that this is when we are discussing propositional logic

• $\equiv$: Logical equivalence
• $\leftrightarrow$: Material equivalence (also called bi-implication)

This can be proved from T1 (Semantic Consequences) and the definition of bi-implication