Biconditional (if-and-only-if)
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.
The biconditional is a complex operator, built out of simpler operators. Think of it this way:
- 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 ).
- ( If P then Q ) and ( P only if Q ). This is like saying:
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.