# 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 |