# Denying the antecedent

A formal logical fallacy in which a negative condition (consequent) is (incorrectly) inferred from a negative condition (antecendent).

For example:

If A lives in London, [then] A lives in England.
A does not live in London.
Therefore, A does not live in England.

Even if the premises hold true, one cannot conclude that from a negative of the condition follows a negative consequence. In this example: There are other places in England where A could live.

## Explanation

This fallacy comes from incorrectly applying the modus tollens and/or modus ponens, or when a conditional is confused with a biconditional.

For comparison, the following table contrasts common valid forms of inference with the fallacy:

## Origin of the term

In a logical conditional, i.e. a statement of the form “if A, then B” (`A → B`), we call A the antecedent (or condition), and B consequent (also consequence).

The name indicates that, in contrast to the (valid) modus tollens, it is not the consequent that is negated but the antecedent, which leads to an invalid conclusion.

## When are such inferences valid?

Denying the antecedent is an invalid conclusion for conditional statements. However, it is explicitly valid for bi­conditionals, which in turn are a special case of conditionals.

Thus, if it can be proved that in addition to `A → B`, also `A ≡ B` is valid, it follows that `B → A` and we have a valid modus tollens.