# modus tollens

The law of *modus tollens ^{}* is the inference rule which allows one to
conclude $\mathrm{\neg}P$ from $P\Rightarrow Q$ and $\mathrm{\neg}Q$. The name “modus
tollens” refers to the fact that this rule allows one to take away the
conclusion

^{}of a conditional

^{}statement and conclude the negation

^{}of the condition. As an example of this rule, we may cite the following:

$$\frac{\genfrac{}{}{0pt}{}{\text{If the postman is at the door, the doorbell will ring twice}}{\text{The bell is not ringing.}}}{\text{The postman is not at the door.}}$$ |

The validity of this rule may be established by means of the following
truth table^{}:

$P$ | $Q$ | $P\Rightarrow Q$ | $\mathrm{\neg}P$ | $\mathrm{\neg}Q$ |
---|---|---|---|---|

F | F | T | T | T |

F | T | T | T | F |

T | F | F | F | T |

T | T | T | F | F |

This rule can be used to justify the popular technique of proof by
contradiction^{}. In this technique, one assumes a hypothesis^{} $P$ and
then derives a conclusion $Q$. This is tantamount to showing that
$P\Rightarrow Q$. Next one demonstrates $\mathrm{\neg}Q$. Applying modus
tollens, one then concludes $\mathrm{\neg}P$.

Title | modus tollens |
---|---|

Canonical name | ModusTollens |

Date of creation | 2013-03-22 16:56:03 |

Last modified on | 2013-03-22 16:56:03 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 03B22 |

Classification | msc 03B35 |

Classification | msc 03B05 |