Modus ponendo tollens

Overview

MPT 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. ${\displaystyle \neg (A\land B)}$
2. ${\displaystyle A}$
3. ${\displaystyle \therefore \neg B}$

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

1. ${\displaystyle A\,|\,B}$
2. ${\displaystyle A}$
3. ${\displaystyle \therefore \neg B}$

Formal Proof

Step Proposition Derivation
1${\displaystyle \neg (A\land B)}$Given
2${\displaystyle A}$Given
3${\displaystyle \neg A\lor \neg B}$De Morgan's laws (1)
4${\displaystyle \neg (\neg A)}$Double negation (2)
5${\displaystyle \neg B}$Disjunctive syllogism (3,4)