Modus ponendo tollens

Transformation rules
Propositional calculus
Rules of inference
Modus ponens / Modus tollens Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / Hypothetical syllogism Constructive / Destructive dilemma Absorption Modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

Modus ponendo tollens (Latin: "mode that by affirming, denies")^[1] is a valid rule of inference for propositional logic, sometimes abbreviated MPT.^[2] It is closely related to modus ponens and modus tollens. It 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:

$\neg (A \and B)$
$A\,\!$
$\therefore \neg B$

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

$A\,|\,B$
$A\,\!$
$\therefore \neg B$

References

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

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