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# modus tollens

The law of *modus tollens* is the inference rule which allows one to
conclude $\neg P$ from $P\Rightarrow Q$ and $\neg Q$. 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:

${{\hbox{If the postman is at the door, the doorbell will ring twice}\atop\hbox% {The bell is not ringing.}}\over\hbox{The postman is not at the door.}}$ |

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

$P$ | $Q$ | $P\Rightarrow Q$ | $\neg P$ | $\neg Q$ |
---|---|---|---|---|

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 $\neg Q$. Applying modus tollens, one then concludes $\neg P$.

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## Mathematics Subject Classification

03B22*no label found*03B35

*no label found*03B05

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