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

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)

Strong form

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

1. ${\displaystyle A{\underline {\lor }}B}$
2. ${\displaystyle A}$
3. ${\displaystyle \therefore \neg B}$

See also

• Modus tollendo ponens
• Stoic logic

References

1. Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.
2. Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London: Routledge. p. 60. ISBN 0-415-91775-1.
3. Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.

• Latin logical phrases
• Rules of inference
• Theorems in propositional logic