# 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

 $a\rightarrow b$

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

a b $a\rightarrow b$
F F T
F T T
T F F
T T T
 Title implication Canonical name Implication Date of creation 2013-03-22 11:53:00 Last modified on 2013-03-22 11:53:00 Owner akrowne (2) Last modified by akrowne (2) Numerical id 10 Author akrowne (2) Entry type Definition Classification msc 03B05 Classification msc 81T70 Classification msc 81T60 Synonym conditional truth function Related topic PropositionalLogic Defines vacuously true Defines implies