Denying the antecedent
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Denying the antecedent is a formal logical fallacy that confuses the directionality of logical relationships. The name derives from ignoring (denying) the "if" statement (the antecendent) in the formal logic and confusing it with the effects of an "if-and-only-if" statement.
The general form of the fallacy is as follows:
- If P, then Q.
- Not P.
- Therefore, not Q.
Because the logical rules laid out don't state that Q is exclusively a condition of P, it is erroneous to assume Q is not present if P is not. There are potentially other factors that could lead to Q that are not solely dependent upon, or indicated by, P.
(The correct implication is the contrapositive: If P, then Q. Not Q, therefore not P.)
The fallacy can be shown with an example with an obviously false conclusion:
- 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.
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, but one can be alive and not hopping.
 Proper form
To avoid the fallacy, the first condition needs to be specifically modified with an "if-and-only-if" clause (iff). This makes the conclusion ¬P therefore ¬Q valid. For a example:
- I can get into this safe if, and only if, I have the keys.
- I don't have the keys.
- I can't get into the safe!
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 valid one. It's easy to change to an if statement, but sometimes that might just be arguing by definition—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 chronology, can be valid if it contradicts the scriptural record."
The misapplication of this can crop up in poor interpretations of medical research, especially those reported in newspapers, 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 causes means you will avoid cancer too.