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Denying the antecedent (also known asdenial of the antecedent,inverse error, orfallacy of the inverse) is aformal fallacy of inferring theinverse from an original statement. Phrased another way, denying the antecedent occurs in the context of anindicative conditional statement and assumes that the negation of theantecedent implies the negation of theconsequent. It is a type of mixedhypothetical syllogism that takes on the followingform:[1]
which may also be phrased as
Arguments of this form areinvalid. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.
The namedenying theantecedent derives from the premise "notP", which denies the "if" clause (antecedent) of theconditional 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 under the logic ofmodus tollens.
A related fallacy isaffirming the consequent. Two relatedvalid forms of logical arguments includemodus ponens (affirming the antecedent) andmodus tollens (denying the consequent).
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:
That argument is intentionally bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example offered byAlan Turing in the article "Computing Machinery and Intelligence":
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:
[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.]
