Biconditionals
Converse: The implication \(Q \implies P\) is called the converse of the implication \(P \implies Q\).
- True for open sentences too!
Note!
Note!
\(P \implies Q\) and \(P \implies Q\) have no logical connection!
The Double Implication (biconditional)
\[ P \iff Q \overset{def}{=} (P \implies Q) \ \wedge (Q \implies P) \]
We say:
- \(P \iff Q\)
- \(P\) is equivalent to \(Q\)
- \(P\) if and only if \(Q\)
- \(P\) iff \(Q\)
- Older: \(P\) is necessary and sufficient for \(Q\)
Whenever you're given an English phrasing for double implication, immediately replace it by \(\iff\) and proceed from there.
IMPORTANT: \(P \iff Q = T\) for \(P, Q = (T, T)\) OR \(P, Q = (F, F)\)