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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.
Overview

MPT is usually described as having the form:

Not both A and B
A
Therefore, not B

For example:

Ann and Bill cannot both win the race.
Ann won the race.
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:

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

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

\( A\,|\,B\)
A
\( \therefore \neg B\)


|}
See also

Modus tollendo ponens
Stoic logic

References

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

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