Denying the Antecedent


Sibling Fallacy: Affirming the Consequent

Alias: Denial of the Antecedent1

Form Similar Validating Forms
Modus Ponens Modus Tollens
If p then q.
Therefore, not-q.
If p then q.
Therefore, q.
If p then q.
Therefore, not-p.






If you behead the King, then he will die.
You won't behead the King.
Therefore, the King won't die.


A conditional statement is a propostion of the "if...then..." form, for instance: "If today is Tuesday then I have logic class." The antecedent of such a statement is the component proposition following "if". In the example, the antecedent is: "Today is Tuesday." To deny the antecedent, of course, is to claim that it is false; to deny the antecedent of the example is to claim: "Today is not Tuesday."

In an argument of the form of denying the antecedent―see the Form in the table, above―the conclusion denies the consequent of the conditional statement, that is, the propositional component following "then". In the example, the consequent is "I have logic class", and its denial is "I don't have logic class." Putting it all together, denying the antecedent is a form of argument with a conditional premiss, another premiss that denies the antecedent of the conditional premiss, and a conclusion that denies its consequent.

Denying the antecedent is a non-validating form of argument because from the fact that a sufficient condition for a statement is false one cannot validly conclude the statement's falsity, since there may be another sufficient condition which is true. For instance, from the fact that it isn't raining, we cannot infer with certainty that the streets are not wet, since they may have been recently washed. Also see the Counter-Example, above, which is an invalid argument of the form of the fallacy, which shows that the form is not validating. While decapitation is a sufficient condition for death, the King will die sometime anyway even if he is not beheaded. Also, if I have logic class on both Tuesday and Thursday, then it doesn't follow from the fact that today is not Tuesday that I don't have logic class, since today might be Thursday.


As mentioned in the Exposition section, above, the form of Denying the Antecedent is non-validating, which means that not every argument of that form is valid. This doesn't mean that every argument that denies the antecedent is invalid; rather, it means that some arguments of that form are invalid. There are arguments of that form that are formally valid, but all of them are such that the second premiss alone implies the conclusion, that is, the immediate inference from the second premiss to the conclusion is valid. Therefore, before pronouncing an instance of denying the antecedent invalid, check to see whether the second premiss implies the conclusion.




Analysis of the Example:


You should not believe in either atheism or pantheism.
Therefore, you should not believe in evolution.



  1. A. R. Lacey, A Dictionary of Philosophy (Third Revised Edition) (Barnes & Noble, 1996).