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 truth table for Not PQ is as follows:

If you have Java, try it out for yourself:

Set the truth values for the propositions P and Q by clicking on the appropriate button beside "P" and "Q" respectively. What is the truth value of P id and only if Q? Find out by clicking on the "Compute" button.

P is: T F Q is: T F
Thus, P if and only if Q is:
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 ).
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.

17 August 1996