Modus ponendo tollens

Rules of inference
Propositional calculus
Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) Modus ponendo tollens (¬(A∧B), A ⊢ ¬B) Conjunction introduction (A, B ⊢ A∧B) Simplification (A∧B ⊢ A) Disjunction introduction (A ⊢ A∨B) Disjunction elimination (A∨B, A→C, B→C ⊢ C) Disjunctive syllogism (A∨B, ¬A ⊢ B) Hypothetical syllogism (A→B, B→C ⊢ A→C) Constructive dilemma (A→P, B→Q, A∨B ⊢ P∨Q) Destructive dilemma (A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B) Biconditional introduction (A→B, B→A ⊢ A↔B) Biconditional elimination (A↔B ⊢ A→B)
Predicate calculus
Universal generalization Universal instantiation Existential generalization Existential instantiation
v · d · e

Modus ponendo tollens (Latin: mode that by affirming, denies)^[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 will win the race.
3. Therefore, 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. $\neg (A \and B)$
2. A
3. $\therefore \neg B$

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.