Affirming the consequent

Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dark,") and invalidly inferring its converse ("The room is dark, so the lamp is broken,") even though the converse may not be true. This arises when a consequent ("the room would be dark") has one or more other antecedents (for example, "the lamp is not plugged in" or "the lamp is in working order, but is switched off").

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.

Formal Description

Affirming the consequent is the action of taking a true statement $P\to Q$ and invalidly concluding its converse $Q\to P$. The name affirming the consequent derives from using the consequent, Q, of $P\to Q$, to conclude the antecedent P. This illogic can be summarized formally as $(P\to Q,Q)\to P$ or, alternatively, ${\frac {P\to Q,Q}{\therefore P}}$.

The root cause of such a logic 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. $P\leftrightarrow Q$, it is possible to infer P under the condition Q. For example, the statements "It is August 13, so it is my birthday" $P\to Q$ and "It is my birthday, so it is August 13" $Q\to P$ are equivalent and both true consequences of the statement "August 13 is my birthday" (an abbreviated form of $P\leftrightarrow Q$). Using one statement to conclude the other is not an example of affirming the consequent, but some person misapply the approach.

It may be worth noting that $P\to Q$ does imply its contrapositive, $\neg Q\to \neg P$, where $\neg Q,\neg P$ symbolize the negations of Q and P, respectively. For example, the statement "If the lamp were broken, then the room would be dark," ($P\to Q$) does imply its contrapositive, "The room is not dark, so the lamp is not broken," ($\neg Q\to \neg P$).

Additional Examples

Example 1

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 Bill Gates owns Fort Knox, then Bill Gates is rich.

Bill Gates is rich.

Therefore, Bill Gates owns Fort Knox.

Owning Fort Knox is not the only way to be rich. Any number of other ways to be rich exist.

However, one can affirm with certainty that "if someone is not rich" (non-Q), then "this person does not own Fort Knox" (non-P). This is the contrapositive of the first statement, and it must be true if and only if the original statement is true.

Example 2

Here is another useful, obviously-fallacious example, but one that does not require familiarity with who Bill Gates is and what Fort Knox is:

If an animal is a dog, then it has four legs.

My cat has four legs.

Therefore, my cat is a dog.

Here, it is immediately intuitive that any number of other antecedents ("If an animal is a deer...", "If an animal is an elephant...", "If an animal is a moose...", etc.) can give rise to the consequent ("then it has four legs"), and that it is preposterous to suppose that having four legs must imply that the animal is a dog and nothing else. This is useful as a teaching example since most people can immediately recognize that the conclusion reached must be wrong (intuitively, a cat cannot be a dog), and that the method by which it was reached must therefore be fallacious.

Example 3

Arguments of the same form can sometimes seem superficially convincing, as in the following example:

If Brian had been thrown off the top of the Eiffel Tower, then he would be dead.

Brian is dead.

Therefore, Brian was thrown off the top of the Eiffel Tower.

Being thrown off the top of the Eiffel Tower is not the only cause of death, since there exist numerous different causes of death.

Affirming the consequent is commonly used in rationalization, and thus appears as a coping mechanism in some people.

Example 4

In Catch-22, the chaplain is interrogated for supposedly being 'Washington Irving'/'Irving Washington', who has been blocking out large portions of soldiers' letters home. The colonel has found such a letter, but with the Chaplain's name signed.

'You can read, though, can't you?' the colonel persevered sarcastically. 'The author signed his name.'

'That's my name there.'

'Then you wrote it. Q.E.D.'

P in this case is 'The chaplain signs his own name', and Q 'The chaplain's name is written'. The chaplain's name may be written, but he did not necessarily write it, as the colonel falsely concludes.

See also

• Appeal to consequences
• Confusion of the inverse
• Denying the antecedent
• ELIZA effect
• Fallacy of the single cause
• Fallacy of the undistributed middle
• Inference to the best explanation
• Modus ponens
• Modus tollens
• Post hoc ergo propter hoc
• Necessity and sufficiency