biconditionalBiconditional2001-10-24 20:34:282006-10-01 23:16:56Definition\usepackage{amssymb}
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A \emph{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
\begin{center}
\begin{tabular}{ccc}
a & b & $a \Leftrightarrow b$ \\
\hline
F & F & T \\
F & T & F \\
T & F & F \\
T & T & T
\end{tabular}
\end{center}
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.
\section{Colloquial Usage}
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 \emph{not} meant as a biconditional, since it can obviously be cloudy while not raining.