Modus ponendo tollens
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Modus ponendo tollens (MPT;Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234. Latin: "mode that denies by affirming"){{cite book |last=Stone |first=Jon R. |year=1996 |title=Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language |location=London |publisher=Routledge |isbn=0-415-91775-1 |page=[https://archive.org/details/latinforillitera0000ston/page/60 60] |url=https://archive.org/details/latinforillitera0000ston |url-access=registration }} 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."Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.
In logic notation this can be represented as:
Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:
Proof
class="wikitable"
! Step ! Proposition ! Derivation | ||
1 | Given | |
2 | Given | |
3 | De Morgan's laws (1) | |
4 | Double negation (2) | |
5 | Disjunctive syllogism (3,4) |
Strong form
Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise: