Denying the antecedent

Denying the antecedent (also fallacious modus tollens) is a formal fallacy that confuses the directionality of logical relationships. The name derives from ignoring (denying) the "if" statement (the antecedent) in the formal logic and confusing it with the effects of an "if-and-only-if" statement.

The fallacy is a causation fallacy and a formal fallacy.

Form
The general form of the fallacy is as follows:

Because the logical rules laid out don't state that Q is exclusively a condition of P, it is incorrect to assume Q is not present if P is not. In most cases, there are other reasons that Q could be false.

This argument would only be valid if it is true that "if and only if A is true, B is true", which is normally not the case.

(The correct implication is the contrapositive: If P, then Q. Not Q, therefore not P.)

Example
The fallacy can be shown using an example with an obviously false conclusion:

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, either the condition needs to be specifically modified with an "if-and-only-if" clause (iff) or the conclusion needs to be specifically modified with a “probably” clause. This makes the conclusion ¬P therefore (probably) ¬Q valid.

Examples:

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 sound 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 the media, 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 the causes means you will avoid cancer too.