#
*Modus ponendo tollens*

* Modus ponendo tollens* (

**MPT**;[1] Latin: "mode that denies by affirming")[2] is a valid rule of inference for propositional logic. It is closely related to

*modus ponens*and

*modus tollendo ponens*.

Transformation rules |
---|

Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

## Overview

MPT is usually described as having the form:

- Not both A and B
- A
- Therefore, not B

For example:

- Ann and Bill cannot both win the race.
- Ann won the race.
- Therefore, Bill cannot have won the race.

As E. J. Lemmon describes it:"*Modus ponendo tollens* is the principle that, if the negation of a conjunction holds and also one of its conjuncts, then the negation of its other conjunct holds."[3]

In logic notation this can be represented as:

Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:

## Formal Proof

Step |
Proposition |
Derivation |
---|---|---|

1 | Given | |

2 | Given | |

3 | De Morgan's laws (1) | |

4 | Double negation (2) | |

5 | Disjunctive syllogism (3,4) |

## See also

## References

- Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'.
*Thinking and Reasoning*. 7:217–234. - Stone, Jon R. (1996).
*Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language*. London: Routledge. p. 60. ISBN 0-415-91775-1. - Lemmon, Edward John. 2001.
*Beginning Logic*. Taylor and Francis/CRC Press, p. 61.

This article is issued from
Wikipedia.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.