Modus ponendo tollens

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.

Transformation rules
Propositional calculus
Rules of inference
Rules of replacement
Predicate logic

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:

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


|}

See also

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.
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