Any two propositions P and Q can be joined with the biconditional operator, producing the new, complex, proposition:

P if and only if Q

The proposition P if and only if Q is true if and only if both P and Q are true, or if both P and Q are false. It is false only when one of them is true and the other false.

Examples

P	Q	Not P if and only if Q
T	T	T
T	F	F
F	T	F
F	F	T

Proof

The biconditional is a complex operator, built out of simpler operators. Think of it this way:

<dl>

P if and only if Q is the same as:

( If P then Q ) and ( P only if Q ). This is like saying:

( If P then Q ) and ( If Q then P ). </dl>

The if and only if operator plays a special role in definitions. When we say P if and only if q, we are saying that P says the same thing as Q.