Affirming the consequent

1. If P, then Q.
2. Q.
3. Therefore, P.

which may also be phrased as

1. ${\displaystyle P\to Q}$ (P implies Q)
2. ${\displaystyle \therefore Q\to P}$ (therefore, Q implies P)

For example, it may be true that a broken lamp would cause a room to become dark. It is not true, however, that a dark room implies the presence of a broken lamp. There may be no lamp (or any light source), or the lamp might be functional but switched off. In other words, the consequent (a dark room) can have other antecedents (no lamp, off-lamp), and so can still be true even if the stated antecedent is not.

Converse errors are common in everyday thinking and communication and can result from, among other causes, communication issues, misconceptions about logic, and failure to consider other causes.

A related fallacy is denying the antecedent. Two related valid forms of logical argument include modus tollens (denying the consequent) and modus ponens (affirming the antecedent).

Formal description

Affirming the consequent is the action of taking a true statement ${\displaystyle P\to Q}$ and invalidly concluding its converse ${\displaystyle Q\to P}$. The name affirming the consequent derives from using the consequent, Q, of ${\displaystyle P\to Q}$, to conclude the antecedent P. This fallacy can be summarized formally as ${\displaystyle (P\to Q,Q)\to P}$ or, alternatively, ${\displaystyle \frac{P\to Q,Q}{\therefore P}}$. The root cause of such a logical error is sometimes failure to realize that just because P is a possible condition for Q, P may not be the only condition for Q, i.e., Q may follow from another condition as well.

Affirming the consequent can also result from overgeneralizing the experience of many statements having true converses. If P and Q are "equivalent" statements, i.e., ${\displaystyle P\leftrightarrow Q}$, it is possible to infer P under the condition Q. For example, the statements "It is August 13, so it is my birthday" ${\displaystyle P\to Q}$ and "It is my birthday, so it is August 13" ${\displaystyle Q\to P}$ are equivalent and both true consequences of the statement "August 13 is my birthday" (an abbreviated form of ${\displaystyle P\leftrightarrow Q}$).

Of the possible forms of "mixed hypothetical syllogisms", two are valid and two are invalid. Affirming the antecedent (modus ponens) and denying the consequent (modus tollens) are valid. Affirming the consequent and denying the antecedent are invalid.