# Modus ponendo tollens

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:

1. Not both A and B
2. A
3. Therefore, not B

For example:

1. Ann and Bill cannot both win the race.
2. Ann won the race.
3. 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:

1. $\neg (A \and B)$
2. $A\,\!$
3. $\therefore \neg B$

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

1. $A\,|\,B$
2. $A\,\!$
3. $\therefore \neg B$

## References

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

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