Modus ponendo tollens

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Modus ponendo tollens (MPT;Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234. Latin: "mode that denies by affirming"){{cite book |last=Stone |first=Jon R. |year=1996 |title=Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language |location=London |publisher=Routledge |isbn=0-415-91775-1 |page=[https://archive.org/details/latinforillitera0000ston/page/60 60] |url=https://archive.org/details/latinforillitera0000ston |url-access=registration }} 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."Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.

In logic notation this can be represented as:

  1. \neg (A \land B)
  2. A
  3. \therefore \neg B

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

  1. A\,|\,B
  2. A
  3. \therefore \neg B

Proof

class="wikitable"

! Step

! Proposition

! Derivation

1\neg (A \land B) Given
2AGiven
3\neg A \lor \neg BDe Morgan's laws (1)
4\neg \neg ADouble negation (2)
5\neg BDisjunctive syllogism (3,4)

Strong form

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

  1. A \underline\lor B
  2. A
  3. \therefore \neg B

See also

References