Affirming the consequent

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

Contents

[edit] Forms of the argument

[edit] Propositional logic

  • If P, then Q.
  • Q.
  • Therefore, P.

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

[edit] First-order logic

  • If P is true of x, then Q is true of x.
  • Q is true of x.
  • Therefore, P is true of x.

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

[edit] Fallaciousness

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.

[edit] Examples of the argument

[edit] Propositional form

  • If it is sunny today, then I will go swimming.
  • I will go swimming.
  • Therefore, it is sunny today.


  • If Bill Gates owns Fort Knox, then he is rich.
  • Bill Gates is rich.
  • Therefore, Bill Gates owns Fort Knox.


  • If the Bible is true, then Jerusalem is a real city.
  • Jerusalem is a real city.
  • Therefore the Bible is true.


  • If God answers prayers, then the tumor will be benign.
  • The tumor is benign.
  • Therefore, God answers prayers.


[edit] First-order form

  • Fascists support a strong military.
  • John Q. Warhawk supports a strong military.
  • Therefore, John Q. Warhawk is a fascist.

[edit] See also

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