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→b |
or
a⇒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 (⇔). They are not, however, the same. The implication a→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→b is called vacuously true when a is false. By contrast, a⇔b would be false if either a or b was by itself false (a⇔b≡(a∧b)∨(¬a∧¬b), or in terms of implication as (a→b)∧(b→a)).
It may be useful to remember that a→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→b is in fact equivalent to
b∨¬a |
The truth table for implication is therefore
a | b | a→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 |