Modus ponendo tollens explained

Modus ponendo tollens (MPT;^[1] Latin: "mode that denies by affirming")^[2] 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."^[3]

In logic notation this can be represented as:

\neg (A \land B) A \therefore \neg B

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

A|B A \therefore \neg B

Proof

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

Strong form

Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:

A \underline\lor B A \therefore \neg B

See also

• Modus tollendo ponens
• Stoic logic

Notes and References

1. Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.
2. Book: Stone, Jon R. . 1996 . Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language . London . Routledge . 0-415-91775-1 . 60 . registration .
3. [E. J. Lemmon|Lemmon, Edward John]