Implications 'If and Only If'

When a statement is true in both directions — if $P$ then $Q$ and if $Q$ then $P$ — we say $P$ holds if and only if $Q$ holds. This two-way logical connection is written as $P \iff Q$, and it means $P$ and $Q$ are logically equivalent: one cannot be true without the other. Understanding this biconditional is crucial because it captures the strongest kind o …