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