# Affirming the consequent

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Affirming the consequent is a logical fallacy confusing the directionality of if-then propositions, and named after the consequent in the conditional statement (Q in "if P, then Q").

The fallacy is a formal fallacy.

## Forms of the argument

P1: If X, then Y.
P2: Y.
C1: X.

In formal terms, the fallacious argument is stated as $\left(P\rightarrow Q\right), Q \vDash P$.

### Converting a conditional

Converting a conditional occurs when the components of a compound if statement are switched.

P1: If X, then Y.
C1: If Y, then X.

## Explanation

Affirming the consequent related to the generic phrase that "all x are y, but not all y are x" in that the formal fallacy fails to recognise the "not all y are x" part. Its statistical equivalent is confusion of the inverse, where two conditional probabilities are mistaken to be equal when this is not necessarily true.

As a formal fallacy, it is an error in the underlying logic of an argument or proposition and, similar to how denying the antecedent can be remedied, is corrected by replacing the directional condition "if" with the bidirectional equivalence of the "if and only if" statement. This would mean that P necessitates Q and, equally, Q necessitates P. While this will correct the formal logic and the hypothetical assertions made, it can still form a not even wrong argument if the "if and only if" premise happens to be not well founded.

## Examples

P1: If the Bible is true, then Jerusalem is a real city.
P2: Jerusalem is a real city.
C1: Therefore the Bible is true.

P1: If the universe was created by God, we would see order everywhere.
P2: And we do see order, not randomness.
C1: It's clear that the universe had a creator.

P1: If it is sunny today, then I will go swimming.
P2: I will go swimming.
C1: Therefore, it is sunny today.

P1: If Bill Gates owns Fort Knox, then he is rich.
P2: Bill Gates is rich.
C1: Therefore, Bill Gates owns Fort Knox.

P1: If God answers prayers, then the tumor will be benign.
P2: The tumor is benign.