Affirming the consequent

{{logicnav}}
'''Affirming the consequent''' (or '''fallacious [[modus ponens]]''') is a [[logical fallacy]] confusing the directionality of if-then propositions, and named after the ''consequent'' in the conditional statement ('''Q''' in "if '''P''', then '''Q'''"). 

The fallacy is a [[formal fallacy]].

==Form==
{{syl
|2=If '''X''', then '''Y'''.
|4='''Y'''.
|5=C|6='''X'''.
}}

In formal terms, the fallacious argument is stated as <math>\left(P\rightarrow Q\right), Q \vDash P</math>.

==Variants==

===Converting a Conditional===
Converting a conditional occurs when the components of a compound if statement are switched.
{{syl
|1=P|2=If '''X''', then '''Y'''.
|3=C|4=If '''Y''', then '''X'''.
}}

===False Conversion===
This assumes that since all members of one group are part of a second group, all members of the second group must be part of the first group.
{{syl
|1=P|2=All '''X''' are '''Y'''.
|3=C|4=Therefore, all '''Y''' are '''X'''.
}}

'''Examples:'''
{{syl
|1=P|2=All dogs are animals.
|3=C|4=Therefore, all animals are dogs.
}}

{{syl
|1=P|2=All [[Bible thumper]]s are Christians.
|3=C|4=Therefore, all Christians are Bible thumpers.
}}

{{syl
|1=P|2=All [[Communism|communists]] are leftists.
|3=C|4=Therefore, all leftists are communists.
}}

{{syl
|1=P|2=All [[Neo-Nazism|Neo-Nazis]] are right-wing.
|3=C|4=Therefore, all right-wingers are Neo-Nazis.
}}

{{syl
|1=P|2=All squares are rhombuses.
|3=C|4=Therefore, all rhombuses are squares.
}}

===Kafkatrapping===

This is where a person's denial of being involved in a wrongdoing is used as further evidence of that wrongdoing taking place. These kinds of [[Falsifiability|unfalsifiable]] accusations were commonplace during the times of [[McCarthyism]] and the [[witch hunt]]s, and are especially popular among [[Conspiracy theory|conspiracy theorists]].

The form goes as follows: 
{{syl
|2=If a person is guilty of a wrongdoing, they will deny being guilty of it.

|4=Person X denies being guilty.

|5=C|6=Therefore, person X is guilty.
}}

Pointing out that guilty people usually plead innocent only shows that the accused ''could'' theoretically be guilty, not that they are. As always, the [[burden of proof]] lies on the person making the claim.

===If and only if===

Within logic the bi-conditional variant of affirming the consequent is technically non-fallacious (though it can be argued that it is not truly affirming the consequent to begin with).  This takes the form of.... 
{{syl
|2='''A''' If and only if '''B'''.
|4='''B'''.
|5=C|6= Therefore '''A'''.
}}

This is valid because the phrase "'''A''' if and only if '''B'''" is really equivalent to "If '''A''' then '''B''' ''and'' If '''B''' then '''A'''" .  So the "consequent" in this case is also a type of antecedent making the inference one of [[modus ponens]].

==Explanation==
Affirming the consequent is related to the generic phrase that "all ''X'' are ''Y'', but not all ''Y'' are ''X''" in that the formal fallacy fails to recognise the "not all ''Y'' are ''X''" part. Its [[statistical]] equivalent is [[confusion of the inverse]], where two conditional [[probability|probabilities]] are mistaken to be equal when this is not necessarily true.

As a formal fallacy, it is an error in the underlying logic of an argument or proposition and, similar to how [[denying the antecedent]] can be remedied, is corrected by replacing the directional condition "if" with the bidirectional equivalence of the "if and only if" statement or softening the conclusion to assert P as merely probable given Q. This would mean that '''P''' necessitates '''Q''' and, equally, '''Q''' necessitates '''P'''. While this will correct the formal logic and the hypothetical assertions made, it can still form a [[not even wrong]] argument if the "if and only if" premise happens to be not well founded.

==Examples==
{{syl
|2=[[Spider-man fallacy|If the Bible is true, then Jerusalem is a real city.]]
|4=[[Jerusalem]] is a real city.
|5=C|6=Therefore the [[Bible]] is probably true.
}}

{{syl
|2=If [[reality]] were real, any given one of us would have a statistically significant confidence that they existed.
|4=And I do have statistically significant confidence that I exist.
|5=C|6=It's probably clear that reality is real.
}}

{{syl
|2=If a defendant is guilty, then they will deny being guilty.
|4=The defendant denied being guilty.
|5=C|6=Therefore, the defendant is guilty.
}}

{{syl
|2=If the [[Universe]] was created by [[God]], we would see order everywhere.
|4=And we do see order, not [[random]]ness, everywhere we look.
|5=C|6=It's probably clear that the universe had a creator (specifically, the locally popular one).<ref group=note>This is an example of a [[transcendental argument for God]]. Transcendental arguments for any other concept follow the same form.</ref>
}}

{{syl
|2=If it is sunny today, then I will go swimming.
|4=I will go swimming.
|5=C|6=Therefore, it is probably sunny today.
}}

{{syl
|2=If [[Bill Gates]] owns [[Fort Knox]], then he is rich.
|4=Bill Gates is rich.
|5=C|6=Therefore, Bill Gates probably owns Fort Knox.
}}

{{syl
|2=(God exists and) If God answers [[prayers]], then the [[tumor]] will be benign.
|4=The tumor is benign.
|5=C|6=Therefore, God probably answers prayers.
}}

{{syl
|2=[[Fascists]] support a strong military.
|4=John Q. Warhawk supports a strong military.
|5=C|6=Therefore, John Q. Warhawk is probably a fascist.
}}

===Conditional===
{{syl
|1=P|2=If educational standards are lowered, argument standards on the internet worsen.
|3=C|4=Therefore, if we see dumber arguments on the internet, we'll probably know that our educational standards are falling.
}}

==See also==
* [[Non sequitur]]
*''[[Post hoc, ergo propter hoc]]''
*[[Causalation]]
*[[Denying the antecedent]]
*[[Modus ponens]]

==External links==
*{{wpa|Affirming the consequent}}
*[http://www.fallacyfiles.org/afthecon.html "Affirming the consequent"], [http://www.fallacyfiles.org/commcond.html "Commuting a Conditional"], [[Fallacy Files]]
*[http://infidels.org/library/modern/mathew/logic.html#consequent "Affirmation of the consequent"],[http://infidels.org/library/modern/mathew/logic.html#conditional "Converting a conditional"], [[Secular Web]]

==Notes==
{{reflist|group=note}}

{{logfal}}

[[Category:Formal fallacies]]

Categories: Formal fallacies