Modus ponendo tollens

Modus ponendo tollens (Latin: "mode that by affirming, denies") is a valid rule of inference for propositional logic, 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. 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."

In logic notation this can be represented as:

1. ¬ ( A ∧ B )
2. A
3. ∴ ¬ B

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

1. A | B
2. A
3. ∴ ¬ B