Denying the antecedent

{{logicnav}}
'''Denying the antecedent''' (also '''fallacious ''[[modus tollens]]''''') is a [[formal fallacy]] that confuses the directionality of logical relationships. The name derives from ignoring (denying) the "if" statement (the antecedent) in the formal logic and confusing it with the effects of an "if-and-only-if" statement.<ref>[http://www.fallacyfiles.org/denyante.html Fallacy Files - Denying the Antecdent]</ref>

The fallacy is a [[causation fallacy]] and a [[formal fallacy]].

==Form==
The general form of the fallacy is as follows:
{{syl
|2=If '''P''', then '''Q'''.
|4=Not '''P'''.
|5=C|6=Therefore, not '''Q'''.
}}

Because the logical rules laid out don't state that '''Q''' is ''exclusively'' a condition of '''P''', it is incorrect to assume '''Q''' is not present if '''P''' is not. In most cases, there are other reasons that '''Q''' could be false.

This argument would only be valid if it is true that "if and only if A is true, B is true", which is normally not the case.

(The correct implication is the contrapositive: If '''P''', then '''Q'''. Not '''Q''', therefore not '''P'''.)

==Example==
The fallacy can be shown using an example with an obviously false conclusion:
{{syl
|1=P1|2=Any person who is hopping on one foot must be alive.
|3=P2|4=A [[sleep]]ing person is not hopping on one foot.
|5=C|6=Therefore, all sleeping people are [[dead]].
}}

In formal logic, the if-then statement (material implication) is often represented by a one-way arrow (→) to indicate that the condition is directional. Meanwhile the iff-then statement (material equivalence, see below) is bidirectional as indicated by the double-headed arrow (↔). The direction means that one property is reliant on the presence of the other, but the same doesn't ''necessarily'' hold true in reverse; hopping on one foot requires being alive{{fact|joke}}, but one can be alive and not hopping.

==Proper form==
To avoid the fallacy, either the condition needs to be specifically modified with an "if-and-only-if" clause (iff) or the conclusion needs to be specifically modified with a “probably” clause.<ref>[http://xkcd.com/1033/ Obligatory xkcd on the "iff" statement]</ref> This makes the conclusion  '''¬P''' therefore (probably) '''¬Q''' [[valid]].

Examples:
{{syl
|2=I can't get into this safe if, and only if, I don't have the keys.
|4=I can't get into the safe!
|5=C|6=I don't have the keys.
}}
{{syl
|2=I can get into this safe if I have the keys.
|4=I can't get into the safe!
|5=C|6=I probably don't have the keys.
}}

This is valid logic, and so there is no formal fallacy here. Of course, formal systems don't take into account the existence of thermite which can easily crack a safe, so the change of the first condition to an if-and-only-if statement may not be a [[Soundness|sound]] one. It's easy to change to an if statement, but sometimes that might just be [[Argumentum ad dictionarium|arguing by definition]]<ref>[http://lesswrong.com/lw/nz/arguing_by_definition/ Less Wrong — Arguing by Definition]</ref> — and anyone can change definitions to whatever they like to prove just about anything, as [[Answers in Genesis]] does when it says that "'''By definition''', no apparent, perceived or claimed evidence in any field, including [[history]] and [[Geologic timeline|chronology]], can be valid if it contradicts the [[Holy book|scriptural]] record."<ref>[http://www.answersingenesis.org/about/faith AIG — Statement of Faith]</ref><ref>[http://www.icr.org/article/does-science-conflict-with-bible/ ICR — Does science conflict with the Bible?]</ref>

The misapplication of this can crop up in poor interpretations of [[medical]] research, especially those reported in the [[media]], where the message often mutates from a [[statistical]] correlation to something like "you will get [[cancer]] if, and only if, you eat red meat / don't drink [[wine]] / use [[Facebook]] / get your medical knowledge from the ''[[Daily Mail]]''." Indeed, denial of the antecedent can appear when these popular media reports also seem to suggest avoiding one or two of the causes means you will avoid cancer too.

==See also==
*[[One single proof]], a subset of this fallacy.
*[[Affirming the consequent]]
*''[[Modus tollens]]''
*''[[Non sequitur]]''

{{otherlang
|ru=Отрицание антецедента}}

==External links==
*[http://www.fallacyfiles.org/denyante.html Denying the Antecedent], [[Fallacy Files]]
*[http://www.iep.utm.edu/fallacy/#DenyingtheAntecedent Denying the Antecedent], Internet Encyclopedia of Philosophy

==References==
{{reflist}}

{{logfal}}

[[Category:Causation fallacies]]
[[Category:Formal fallacies]]

[[ru:Отрицание антецедента]]

Categories: Causation fallacies, Formal fallacies