# Affirming the consequent

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Affirming the consequent, sometimes called converse error, fallacy of the converse or confusion of necessity and sufficiency, is a formal fallacy of inferring the converse from the original statement. The corresponding argument has the general form:

$\frac{P \to Q, Q}{\therefore P}$

An argument of this form is invalid, i.e., the conclusion can be false even when statements 1 and 2 are true. Since P was never asserted as the only sufficient condition for Q, other factors could account for Q (while P was false).

To put it differently, if P implies Q, the only inference that can be made is non-Q implies non-P. (Non-P and non-Q designate the opposite propositions to P and Q.) This is known as logical contraposition. Symbolically:

$(P \to Q)\leftrightarrow (\neg Q \to \neg P)$

The name affirming the consequent derives from the premise Q, which affirms the "then" clause of the conditional premise.

## 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 exist to be rich.

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

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

If I had been thrown off the top of the Eiffel tower, then I am dead.
Therefore, I had been thrown off the top of the Eiffel tower.

But being thrown off the top of the Eiffel tower is not the only cause of death, since there exists numerous different causes of deaths. For instance, one could die from fatal stab woulds, a heart attack, or due to being thrown off a different building (perhaps the Shanghai Tower).

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

Example 3

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 (and in fact he did not, as in the novel, Yossarian signed the name