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A biconditional is a truth function that is true only in the case that both parameters are true or both are false.
Symbolically, the biconditional is written as
$$ a \Leftrightarrow b$$ or $$ a \leftrightarrow b$$
with the latter being rare outside of formal logic. The truth table for the biconditional is
| a |
b |
$a \Leftrightarrow b$ |
| F |
F |
T |
| F |
T |
F |
| T |
F |
F |
| T |
T |
T |
The biconditional function is often written as ``iff,'' meaning ``if and only if.''
It gets its name from the fact that it is really two conditionals in conjunction,
$$ (a \rightarrow b) \land (b \rightarrow a) $$
This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently.
The only unambiguous way of stating a biconditional in plain English is of the form ``$b$ if $a$ and $a$ if $b$ .'' Slightly more formal, one would say ``$b$ implies $a$ and $a$ implies $b$ .'' The plain English ``if'' may sometimes be used as a biconditional. One must weigh context heavily.
For example, ``I'll buy you an ice cream if you pass the exam'' is meant as a biconditional, since the speaker doesn't intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, ``it is cloudy if it is raining'' is not meant as a biconditional, since it can obviously be cloudy while not raining.
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