Modus ponendo tollens

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Modus ponendo tollens (Latin: mode that affirms by denying)[1] is a valid rule of inference, sometimes abbreviated MPT.[2] It is closely related to Modus ponens and modus tollens. It 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. So, Bill cannot win 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. \neg (A \land B)
  2. \ A
  3. \therefore \neg B

Other mathematical and logical symbols may be used to present this same form, such as:

  1. ~(A • B)
  2. A
  3. \therefore ~B

It has also been described as having the following alternative forms:[4]

  1. Either A is B or A is C
  2. A is B
  3. Therefore, A is not C
  1. Either A is B or C is D
  2. A is B
  3. Therefore, C is not D
  1. Either A or B is C
  2. A is C
  3. Therefore, B is not C

[edit] References

  1. ^ Stone, Jon R. 1996. Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge:60.
  2. ^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217-234.
  3. ^ Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press: 61.
  4. ^ Joseph H.W.B. 1950. An Introduction to Logic, 2nd Edition, revised. Oxford: Oxford University Press: 345