Denying the antecedent is a fallacy in formal logic where in a standard if/then premise, the antecedent (what comes after the ‘if’) is made not true, then it is invalidly concluded that the consequent (what comes after the ‘then’) is not true. The fallacy confuses the directionality of logical relationships.
The structure of the fallacy takes the following form:
If P, then Q.
Not P.
Therefore, not Q.
Because this structure does not state that Q is exclusively a condition of P, it is invalid to deduce that Q is not true if P is not true. In most cases, there are other reasons that Q could be false. One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:
If it is raining, then the grass is wet.
It is not raining.
Therefore, the grass is not wet.
The conclusion is invalid because there are other reasons why the grass could be wet at the time (someone could have watered it). To give another example:
Any person who is hopping on one foot, must be alive.
A sleeping person is not hopping on one foot.
Therefore, all sleeping people are dead.
An implication for skepticism is that quacks and conspiracy theorists sometimes use this fallacy in their propaganda, for example:
If pharmaceutical companies were always honest, then we could trust drug X.
Pharmaceutical companies are not always honest,
Therefore, we cannot trust drug X.
Recognising and exposing such a denial of the antecedent could be helpful to skeptics in our debates against our opponents.
The Logical Place 