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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 ).

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.

About Stephen Downes

Many kudos are owed to Stephen Downes, the logician who originally authored this list of fallacies and made it freely available on the Web. His basic structure remains. A mirror of Downes' original site can be found here.