Editing Modus ponendo tollens

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'''''Modus ponendo tollens''''' ('''MPT''';<ref>Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. ''Thinking and Reasoning''. 7:217–234.</ref> [[Latin]]: "mode that denies by affirming")<ref>{{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 }}</ref> is a [[Validity (logic)|valid]] [[rule of inference]] for [[propositional calculus|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."<ref>[[E. J. Lemmon|Lemmon, Edward John]]. 2001. ''Beginning Logic''. [[Taylor and Francis]]/CRC Press, p. 61.</ref>

In [[Table of logic symbols|logic notation]] this can be represented as:
# <math> \neg (A \land B)</math>
# <math> A</math>
# <math> \therefore \neg B</math>

Based on the [[Sheffer Stroke]] (alternative denial), "|", the inference can also be formalized in this way:
# <math> A\,|\,B</math>
# <math> A</math>
# <math> \therefore \neg B</math>

==Proof==
{| class="wikitable"
! ''Step''
! ''Proposition''
! ''Derivation''
|-
| 1 || <math>\neg (A \land B) </math>|| Given
|-
| 2 || <math>A</math> || Given
|-
| 3 || <math>\neg A \lor \neg B</math> || [[De Morgan's laws]] (1)
|-
| 4 || <math>\neg \neg A</math> || [[Double negation]] (2)
|-
| 5 || <math>\neg B</math> || [[Disjunctive syllogism]] (3,4)
|}

==Strong form==
''Modus ponendo tollens'' can be made stronger by using [[exclusive disjunction]] instead of non-conjunction as a premise:
# <math> A \underline\lor B</math>
# <math> A</math>
# <math> \therefore \neg B</math>

==See also==
* ''[[Modus tollendo ponens]]''
* [[Stoic logic]]

==References==
{{Reflist}}

[[Category:Latin logical phrases]]
[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]

[[nl:Modus tollens#Modus ponendo tollens]]