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 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 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