Modus ponendo tollens
Logical rule of inference
Modus ponendo tollens (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollendo ponens.
01Overview
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."
In logic notation this can be represented as:
Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:
02Proof
| Step | Proposition | Derivation |
|---|---|---|
| 1 |  | Given |
| 2 |  | Given |
| 3 |  | De Morgan's laws (1) |
| 4 |  | Double negation (2) |
| 5 |  | Disjunctive syllogism (3,4) |
03Strong form
Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:
Sources and credits
This article is adapted from the Wikipedia article “Modus ponendo tollens”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
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