Modus ponendo tollens

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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.


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. [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]


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


  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.

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