# Affirming the consequent

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Affirming the consequent (or fallacious modus ponens) 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.
C: 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.

P: If X, then Y.
C: 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 or softening the conclusion to assert P as merely probable given Q. 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.
C: Therefore the Bible is probably true.

P1: If reality were real, any given one of us would have a statistically significant confidence that they existed.
P2: And I do have statistically significant confidence that I exist.
C: It's probably clear that reality is real.

P1: If the Universe was created by God, we would see order everywhere.
P2: And we do see order, not randomness, everywhere we look.
C: It's probably clear that the universe had a creator (specifically, the locally popular one).[note 1]

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

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

P1: (God exists and) If God answers prayers, then the tumor will be benign.
P2: The tumor is benign.
C: Therefore, God probably answers prayers.

P1: Fascists support a strong military.
P2: John Q. Warhawk supports a strong military.
C: Therefore, John Q. Warhawk is probably a fascist.

### Conditional

P: If educational standards are lowered, argument standards on the internet worsen.
C: Therefore, if we see dumber arguments on the internet, we'll probably know that our educational standards are falling.