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

a	b	$a\to b$
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