====== Denying the antecedent ======

A [[logic:formal_fallacies:index|formal logical fallacy]] in which a negative condition ([[glossary:consequent|consequent]]) is (incorrectly) inferred from a negative condition ([[glossary:antecedent|antecendent]]).

For example:

> If <var>A</var> lives in London, [//then//] <var>A</var> lives in England.
> <var>A</var> does //not// live in London.
> <span conclusio><s invalid>Therefore, <var>A</var> does not live in England.</s></span>

Even if the premises hold true, one cannot conclude that from a negative of the condition follows a negative consequence. In this example: There are other places in England where <var>A</var> could live.

===== Explanation =====

This fallacy comes from incorrectly applying the <i :la>[[logic:inferences:modus_tollens|modus tollens]]</i> and/or <i :la>[[logic:inferences:modus_ponens|modus ponens]]</i>, or when a [[glossary:conditional|conditional]] is confused with a [[glossary:biconditional|biconditional]].

For comparison, the following table contrasts common valid forms of inference with the fallacy:

<div keep-together print-wide>
| ^  <i :la>[[logic:inferences:modus_tollens|Modus tollens]]</i> \\ (valid inference)  ^  <i :la>[[logic:inferences:modus_ponens|Modus ponens]]</i> \\ (valid inference)  ^ ^  Denying the antecedent  \\ (formal fallacy)  ^
^ Premise 1|  <span "If A, then B">''A → B''</span>  |  <span "If A, then B">''A → B''</span>  | |  <span "If A, then B">''A → B''</span>  |
^ Premise 2|  <span "not B">''⌐B''</span>  |  ''A''  | |  <span "not A">''⌐A''</span>  |
^ Conclusion|  <span "not A">''⌐A''</span>  |  ''B''  | |  <s "not B" invalid short2">''⌐B''</s>  |
</div>

===== Origin of the term =====

In a logical [[glossary:conditional|conditional]], i.e. a statement of the form "if <var>A</var>, then <var>B</var>" (<span "if A, then B">''A → B''</span>), we call <var>A</var> the [[glossary:antecedent|antecedent]] (or condition), and <var>B</var> [[glossary:consequent|consequent]] (also //consequence//).

The name indicates that, in contrast to the (valid) <i :la>[[logic:inferences:modus_tollens|modus tollens]]</i>, it is not the //consequent// that is negated but the //antecedent//, which leads to an invalid conclusion.

===== When are such inferences valid? =====

//Denying the antecedent// is an invalid conclusion for [[glossary:conditional|conditional]] statements. However, it is explicitly valid for [[glossary:biconditional|bi­conditionals]], which in turn are a special case of //conditionals//.

Thus, if it can be proved that in addition to <span "if A, then B">''A → B''</span>, also <span "A exactly when B">''A ≡ B''</span> is valid, it follows that <span "if B, then A">''B → A''</span> and we have a valid <i :la>[[logic:inferences:modus_tollens|modus tollens]]</i>.

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

  * [[logic:formal_fallacies:affirming_the_consequent|Affirming the consequent]]
  * [[glossary:antecedent|Antecedent]]
  * [[glossary:biconditional|Bi­conditional]]
  * <i :la>[[logic:inferences:modus_tollens|Modus tollens]]</i>

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

  * [[https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/77/Denying-the-Antecedent|Denying the Antecedent]] on //Logically Fallacious//
  * [[https://en.wikipedia.org/wiki/Denying_the_antecedent|Denying the Antecedent]] on //Wikipedia//

  * Video: [[https://www.khanacademy.org/partner-content/wi-phi/wiphi-critical-thinking/wiphi-fallacies/v/denying-the-antecedent|Denying the Antecedent]] by //Khan Academy//