Editing Denying the antecedent

{{short description|Logical fallacy}}
{{refimprove|date=December 2016}}
'''Denying the antecedent''' (also known as '''inverse error''' or '''fallacy of the inverse''') is a [[formal fallacy]] of inferring the [[inverse (logic)|inverse]] from an original statement. Phrased another way, denying the antecedent occurs in the context of an [[indicative conditional]] statement and assumes that the negation of the [[Antecedent (logic)|antecedent]] implies the negation of the [[consequent]]. It is a type of mixed [[hypothetical syllogism]] that takes on the following [[argument form|form]]:<ref name="KA">{{Cite web |url=https://www.khanacademy.org/partner-content/wi-phi/wiphi-critical-thinking/wiphi-fallacies/v/denying-the-antecedent |author=Matthew C. Harris |title=Denying the antecedent |publisher=[[Khan academy]]}}</ref>

:If ''P'', then ''Q''.
:Not ''P''.
:Therefore, not ''Q''.

which may also be phrased as

:<math>P \rightarrow Q</math>    (P implies Q)
:<math>\therefore \neg P \rightarrow \neg Q</math>    (therefore, not-P implies not-Q)<ref name= KA/>

[[Argument]]s of this form are [[Validity (logic)|invalid]]. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.

The name ''denying the [[Antecedent (logic)|antecedent]]'' derives from the premise "not ''P''", which denies the "if" clause (antecedent) of the [[indicative conditional|conditional]] premise.

The only situation where one may deny the antecedent would be if the antecedent and consequent represent the same proposition, in which case the argument is trivially valid (and it would [[Begging the question|beg the question]]) under the logic of ''[[modus tollens]]''.

A related fallacy is [[affirming the consequent]]. Two related ''valid'' forms of logical arguments include ''[[modus ponens]]'' (affirming the antecedent) and ''[[modus tollens]]'' (denying the consequent).

== Examples ==
One way to demonstrate the invalidity of this argument form is with an example that has true premises but an obviously false conclusion. For example:

:If you are a ski instructor, then you have a job.
:You are not a ski instructor.
:Therefore, you have no job.<ref name=KA/>

That argument is intentionally bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example offered by [[Alan Turing]] in the article "[[Computing Machinery and Intelligence]]":

:If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.<ref>{{Turing 1950}}</ref>

However, men could still be machines that do not follow a definite set of rules. Thus, this argument (as Turing intends) is invalid.

Another example is:

:If I am [[President of the United States]], then I can veto Congress.
:I am not President.
:Therefore, I cannot veto Congress.
[This is a case of the fallacy denying the antecedent as written because it matches the formal symbolic schema at beginning. The form is taken without regard to the content of the language.]
== See also ==
* [[Affirming the consequent]]
* ''[[Modus ponens]]''
* ''[[Modus tollens]]''
* [[Necessity and sufficiency]]
* [[Plausible reasoning]]

== References ==
{{Reflist}}

== External links ==
* [http://www.fallacyfiles.org/denyante.html FallacyFiles.org: Denying the Antecedent]
* [http://www.safalra.com/philosophy/fallacies/antecedent/ safalra.com: Denying The Antecedent]

{{Formal fallacy}}

[[Category:Formal fallacies]]