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# 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 |

## Mathematics Subject Classification

03B05*no label found*81T70

*no label found*81T60

*no label found*

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## Comments

## implies

I would suggest adding "implies" to the "Other Names" section.