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
editMPT 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
edit| Step | Proposition | Derivation |
|---|---|---|
| 1 | Given | |
| 2 | Given | |
| 3 | De Morgan's laws (1) | |
| 4 | Double negation (2) | |
| 5 | Disjunctive syllogism (3,4) |
Strong form
editModus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:
See also
editReferences
edit- ^ 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.