implicationImplication2001-10-24 14:16:162006-08-02 13:18:15Definitionvacuously trueimplies\usepackage{amssymb}
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\usepackage{xypic}An implication is a logical construction that essentially tells us if one condition is true, then another condition must be also true. Formally it is written $$ a \rightarrow b $$ or $$a \Rightarrow b$$ which would be read ``$a$ implies $b$'', or ``$a$ therefore $b$'', or ``if $a$, then $b$'' (to name a few).
Implication is often confused for ``if and only if'', or the biconditional truth function ($\Leftrightarrow$). They are not, however, the same. The implication $a \rightarrow b$ is true even if only $b$ is true. So the statement ``pigs have wings, therefore it is raining today'', is true if it is indeed raining, despite the fact that the first item is false.
In fact, any implication $a \rightarrow b$ is called \emph{vacuously true} when $a$ is false. By contrast, $a \Leftrightarrow b$ would be false if either $a$ or $b$ was by itself false ($a \Leftrightarrow b \equiv (a \land b) \lor (\lnot a \land \lnot b)$, or in terms of implication as $(a \rightarrow b) \land (b \rightarrow a)$).
It may be useful to remember that $a \rightarrow b$ only tells you that it \emph{cannot} be the case that $b$ is false while $a$ is true; $b$ must ``follow'' from $a$ (and ``false'' does follow from ``false''). Alternatively, $a \rightarrow b$ is in fact equivalent to
$$ b \lor \lnot a $$
The truth table for implication is therefore
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a & b & $a \rightarrow b$ \\
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F & F & T \\
F & T & T \\
T & F & F \\
T & T & T \\
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