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 inferring the converse from the original statement. The corresponding argument has the general form:

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:

${\displaystyle (P\to Q)\leftrightarrow (\neg Q\to \neg P)}$

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

One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:

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.

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