# Modus ponendo tollens

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

* Modus ponendo tollens* (Latin: "mode that by affirming, denies")

^{[1]}is a valid rule of inference for propositional logic, sometimes abbreviated

**MPT**.

^{[2]}It is closely related to

*modus ponens*and

*modus tollens*. It 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:

## References

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

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