Modus ponendo tollens: Difference between revisions
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# <math> A</math> |
# <math> A</math> |
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# <math> \therefore \neg B</math> |
# <math> \therefore \neg B</math> |
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==Formal Proof== |
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%" |
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|+ ''' ''' |
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|- style="background:paleturquoise" |
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! style="width:5%" | ''Step'' |
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! style="width:15%" | ''Proposition'' |
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! style="width:25%" | ''Derivation'' |
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|- |
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| 1 || <math>\neg (A \land B)</math> || Given |
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|- |
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| 2 || <math>A</math> || Given |
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|- |
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| 3 || <math>\neg A \lor \neg B</math> || [[De Morgan's laws]] (1) |
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|- |
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| 4 || <math>\neg \neg A</math> || [[Double negation]] (2) |
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|- |
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| 5 || <math>\neg B</math> || [[Disjunctive syllogism]] (3,4) |
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|} |
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==See also== |
==See also== |
Revision as of 05:38, 24 February 2019
Modus ponendo tollens (MPT;[1] Latin: "mode that denies by affirming")[2] 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."[3]
In logic notation this can be represented as:
Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:
Formal Proof
Step | Proposition | Derivation |
---|---|---|
1 | Given | |
2 | Given | |
3 | De Morgan's laws (1) | |
4 | Double negation (2) | |
5 | Disjunctive syllogism (3,4) |
See also
References
- ^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.
- ^ Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London: Routledge. p. 60. ISBN 0-415-91775-1.
- ^ Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.