Affirming the consequent fallacy { Philosophy Index }

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Affirming the consequent

Affirming the consequent is a logical fallacy, committed by an invalid argument form “If P then Q. Q. Therefore, P”.

The affirming the consequent fallacy may be expressed formally as follows:

α → β, β ∴ α

The argument is invalid because β for some reason other than α.

Affirming the consequent example

If I win the lottery, I will buy a new car.
I will buy a new car.
Therefore, I will win the lottery.

This argument can be demonstrated as invalid by the case that I buy a new car with money that I earned from a lucrative career in philosophy, having failed to win the lottery.

Biconditional version

It should be mentioned that, if the conditional is replaced by a biconditional (if and only if) operator, then the argument becomes valid:

α ↔ β, β ∴ α

In this case, β cannot be true for any reason other than α, so we can infer α's truth from β. For example:

If and only if I win the lottery, I will buy a house of solid gold.
I will buy a house of solid gold.
Therefore, I will win the lottery.

This differs from the affirming the consequent fallacy because it eliminates the possibility of me buying a house of solid gold for any other reason in the first premise, by stating that it will happen if and only if I win the lottery.

A biconditional proof of this nature is essentially a case of affirming the antecedent (or modus ponens) rather than affirming the consequent, since α ↔ β is equivalent to (α → β) ∧ (β → α) [see biconditional introduction rule].