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denying the antecedent
Denying the antecedent (DA) is a formal fallacy, i.e., a logical fallacy that is recognizable by its form rather than its content. DA has the form:
If p then q.
So, not q.
p and q represent different statements. A statement with the form "if p then q" is called a conditional statement. p is called the antecedent and q is called the consequent of the conditional statement.
Arguments with the form DA are fallacious because they are invalid. Being invalid means that their conclusions do not follow from their premises, i.e., it is possible for their premises to be true and their conclusions false. A valid argument is one in which the conclusion follows from its premises, i.e., it is impossible for its premises to be true and its conclusion false.
Below are some examples of the fallacy of denying the antecedent:
If atheism is true, then I'm wasting my time praying for rain.
Atheism is not true.
So, I am not wasting my time praying for rain.
If acupuncture is quack medicine, then sticking people with needles to relieve pain is absurd.
Acupuncture is not quack medicine.
So, sticking people with needles to relieve pain is not absurd.
Or, one might argue:
If sticking people with needles to relieve pain is absurd, then acupuncture is quack medicine.
Sticking people with needles to relieve pain is not absurd.
So, acupuncture is not quack medicine.
If we get greater than chance results from our card-guessing experiment, then telepathy is present.
We did not get greater than chance results from our card-guessing experiment.
So, telepathy is not present.
If he's sweating, then he's not telling the truth.
He not sweating.
So, he's telling the truth.
If the suspect evokes a change in galvanic skin response (from sweating), then he is lying.
The suspect did not evoke a change in galvanic skin response.
Therefore, the suspect is not lying.
In each of the above examples of the fallacy of denying the antecedent the premise of the argument may be true but the conclusion does not follow from the premises. The invalidity of these arguments has nothing to do with their content and is due entirely to their fallacious form. A statement not q never follows from the statements if p then q and not p. Even if the premises of a DA argument are true, the conclusion doesn't follow from them. Being fallacious, however, does not mean that the conclusion is false. For example, the following examples of DC have true conclusions:
If President Obama is Muslim, then he is not a Christian.
He is not a Muslim.
So, President Obama is not not a Christian (i.e., he is a Christian).
If President Obama was born in Nigeria, then he is not an American citizen.
He was not born in Nigeria.
So, President Obama is not not an American citizen (i.e., he is an American citizen).