- Modus ponendo tollens
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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 instantiationModus 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:
- Not both A and B
- A
- Therefore, not B
For example:
- Ann and Bill cannot both win the race.
- Ann will win the race.
- 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:
- A
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.
Categories:- Latin logical phrases
- Rules of inference
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