Modus ponendo tollens

Modus ponendo tollens (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollendo ponens.

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."

In logic notation this can be represented as:

1. [math]\displaystyle{ \neg (A \land B) }[/math]
2. [math]\displaystyle{ A }[/math]
3. [math]\displaystyle{ \therefore \neg B }[/math]

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

1. [math]\displaystyle{ A\,|\,B }[/math]
2. [math]\displaystyle{ A }[/math]
3. [math]\displaystyle{ \therefore \neg B }[/math]

Proof

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

See also

• Modus tollendo ponens
• Stoic logic

References

nl:Modus tollens#Modus ponendo tollens