====== Modus Tollens ======

Abbreviated <abbr :la "Modus Tollens">MT</abbr> or <abbr :la "Modus Tollendo Tollens">MTT</abbr>. One of the most elementary (valid) logical inferences. It is based on a [[glossary:conditional|conditional]] and then denying its [[glossary:consequent|consequent]].

> <span "if A, then B">''A → B''</span>   –   <q>if <var>A</var>, then <var>B</var>.</q>
> <span "not B">''¬B''</span>    –   <q>not <var>B</var>.</q>
> <span conclusio><span "Therefore: not A">''∴ ¬A''</span>    –  <q>therefore: not <var>A</var></q></span>

For example, the following is a valid <i :la>Modus Tollens</i>:

> //If// it is raining, [//then//] the street will be wet.
> The street is //not// wet.
> <span conclusio>//Therefore// it does //not// rain.</span>

===== Name =====

The full name of this inference is "<i :la>modus tollendo tollens</i>". 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: 

  * <i :la>Modus tollendo tollens</i>
  * Denying the consequent
  * <i :la>Negatio consequentiae</i>

===== Fallacies =====

Since the <i :la>Modus Tollens</i> 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 [[logic:inferences:modus_ponens|Modus Ponens]]. This may be the reason why erroneous conclusions based on the <abbr :la "Modus Tollens">MT</abbr> are not uncommon.

The following table compares the <i :la">Modus Tollens</i> and the most important related fallacies:

<div print-wide>
| ^  //Modus tollens// \\ (valid inference)  ^ ^  [[logic:formal_fallacies:affirming the consequent|Affirming the consequent]]  \\ (formal fallacy)  ^  [[logic:formal_fallacies:denying the antecedent|Denying the antecedent]]  \\ (formal fallacy)  ^
^Premise 1 |  <span "if A, then B">A → B</span> \\ <small>(if A, then B</small>  | |  <span "Wenn A, dann B">A → B</span> \\ <small>(if A, then B</small>  |  <span "if A, then B">A → B</span> \\ <small>(if A, then B</small>  | 
^Premise 2 |  <span "not B">⌐B</span> <small>(not B)</small>  | |  B  |  <span "not A">⌐A</span> <small>(not A)</small>  |
^Conclusion |  <span "not A">⌐A</span> <small>(not A)</small>  | |  <s invalid short" "A (invalid)">A</s>  |  <s "not B (invalid)" invalid short2">⌐B</s> <small>(not B)</small>  |
</div>

The fallacies can be summarized as follows:

  * In an [[logic:formal_fallacies:affirming the consequent|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 [[logic:formal_fallacies:denying the antecedent|denying the antecedent]], the negation is retained, but the direction of the inference is reversed, which is not a valid transformation (<span maniculus "more info:">[[paralogisms:illicit_commutation/index|illicit commutation]]</span>).

Both fallacies are described in more detail in the linked articles. 

===== See also =====

  * [[glossary:conditional|(Material) conditional]]
  * [[logic:inferences:destructive_dilemma|Destructive dilemma]]
  * [[logic:inferences:modus_ponens|Modus (ponendo) ponens]]

===== More information =====

  * [[wp>Modus tollens]] on //Wikipedia//

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