Modus ponendo tollens: Difference between revisions
Content deleted Content added
Raycheng200 (talk | contribs) No edit summary |
|||
| Line 25: | Line 25: | ||
# <math> \therefore \neg B</math> |
# <math> \therefore \neg B</math> |
||
==Proof== |
|||
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%" |
|||
|+ ''' ''' |
|||
|- style="background:paleturquoise" |
|||
! style="width:5%" | ''Step'' |
|||
! style="width:15%" | ''Proposition'' |
|||
! style="width:25%" | ''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) |
|||
|} |
|} |
||
==See also== |
==See also== |
||
Revision as of 12:47, 19 February 2021
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:
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.