Denying the antecedent

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Denying the antecedent" – news · newspapers · books · scholar · JSTOR (December 2016) (Learn how and when to remove this message)

Denying the antecedent (also known as inverse error or fallacy of the inverse) is a formal fallacy of inferring the 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 implies the negation of the consequent. It is a type of mixed hypothetical syllogism that takes on the following form:^[1]

If P, then Q.

Not P.

Therefore, not Q.

which may also be phrased as

${\displaystyle P\rightarrow Q}$ (P implies Q)

${\displaystyle \therefore \neg P\rightarrow \neg Q}$ (therefore, not-P implies not-Q)^[1]

Arguments of this form are 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 derives from the premise "not P", which denies the "if" clause (antecedent) of the 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 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).