Modus Tollens
Abbreviated MT or MTT. One of the most elementary (valid) logical inferences. It is based on a conditional and then denying its consequent.
A → B–if A, then B.
¬B–not B.
∴ ¬A–therefore: not A
For example, the following is a valid Modus Tollens:
If it is raining, [then] the street will be wet.
The street is not wet.
Therefore it does not rain.
Name
The full name of this inference is “modus tollendo tollens”. This could be loosely translated as “mode of inferring a denying [statement] by denying [another statement]”.
Other names
In the literature, one may also come across the following alternative names for this inference:
- Modus tollendo tollens
- Denying the consequent
- Negatio consequentiae
Fallacies
Since the Modus Tollens does not necessarily correspond to the colloquial meaning of statements of the type “if … then”, it is usually perceived as less intuitive than, for example, the Modus Ponens. This may be the reason why erroneous conclusions based on the MT are not uncommon.
The following table compares the Modus Tollens and the most important related fallacies:
| Modus tollens (valid inference) | Affirming the consequent (formal fallacy) | Denying the antecedent (formal fallacy) |
||
|---|---|---|---|---|
| Premise 1 | A → B (if A, then B | A → B (if A, then B | A → B (if A, then B |
|
| Premise 2 | ⌐B (not B) | B | ⌐A (not A) | |
| Conclusion | ⌐A (not A) | | |
The fallacies can be summarized as follows:
- In an affirmation of the consequent, the order of the premises is retained, but the (necessary) negation is reversed. As a result, the inference is no longer valid.
- In the case of denying the antecedent, the negation is retained, but the direction of the inference is reversed, which is not a valid transformation (illicit commutation).
Both fallacies are described in more detail in the linked articles.
See also
More information
- Modus tollens on Wikipedia