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# Modus ponendo tollens

Modus ponendo tollens (Latin: mode that denies by affirming) is a valid rule of inference, sometimes abbreviated MPT. 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. So, Bill cannot win 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.

In logic notation this can be represented as:

1. $neg \left(A and B\right)$
2. $A$
3. $therefore neg B$

Other mathematical and logical symbols may be used to present this same form, such as:

1. ~(A • B)
2. A
3. $therefore$ ~B

It has also been described as having the following alternative forms:

1. Either A is B or A is C
2. A is B
3. Therefore, A is not C

1. Either A is B or C is D
2. A is B
3. Therefore, C is not D

1. Either A or B is C
2. A is C
3. Therefore, B is not C

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