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Homemodus tollens

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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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## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

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new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag