implication

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

$$a\to b$$

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 ($\iff$). They are not, however, the same. The implication $a\to b$ 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 $a\to b$ is called vacuously true when $a$ is false. By contrast, $a\iff b$ would be false if either $a$ or $b$ was by itself false ($a\iff b\equiv (a\wedge b)\vee (\mathrm{\neg }a\wedge \mathrm{\neg }b)$, or in terms of implication as $(a\to b)\wedge (b\to a)$).

It may be useful to remember that $a\to b$ 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, $a\to b$ is in fact equivalent to

$$b\vee \mathrm{\neg }a$$

The truth table for implication is therefore