Logical Equivalence

Definition. Two formulas, $P$ and $Q$, are logically equivalent if and only if in all Boolean valuations, $[P]=[Q]$. $P\Lleftarrow \Rrightarrow Q$ which is pronounced “P is logically equivalent to Q”

$P\Lleftarrow \Rrightarrow Q\text{ iff }\models P⟺Q$

Note: Two formulas can be syntactically different stil and logically equivalent.

• If we write P ⇔ Q (substituting some formulas for P and Q) it is a formula that can be T in some BVs and F in others.
• If we write P ⇚⇛ Q (substituting some formulas for P and Q) the formulas P and Q have the same value in all BVs.

Do NOT confuse this with Logical Implication, as Logical Equivalence is much stronger.