# Denying the Antecedent

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Propositional Logic > 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.
Not-p.
Therefore, not-q.
If p then q.
p.
Therefore, q.
If p then q.
Not-q.
Therefore, not-p.

### Example:

…I want to list seventeen summary statements which, if true, provide abundant reason why the reader should reject evolution and accept special creation as his basic world-view. …

14. Belief in evolution is a necessary component of atheism, pantheism, and all other systems that reject the sovereign authority of an omnipotent personal God.2

### Counter-Example:

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

### Exposition:

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.

### Exception:

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.

### Exposure:

Together with its sibling fallacy Affirming the Consequent―see above―this fallacy may result from confusion about the direction of a conditional relation. To deny the consequent of a conditional statement and conclude with the denial of its antecedent is a validating form of argument known as "Modus Tollens"―see the second Similar Validating Form in the table, above. These forms are similar enough that someone might mistakenly confuse one with the other. Similarly, it's possible that someone might confuse denying the antecedent with the validating type of argument known as "Modus Ponens", which has a similar form without the denials―see the first Similar Validating Form in the table, above.

Another possible psychological source for these fallacies is the confusion of a conditional with a biconditional proposition, since an argument of the form of denying the antecedent with a biconditional proposition in place of a conditional one will be valid. For this reason, an argument of the form of denying the antecedent may be an enthymeme, that is, it might have the other direction of the biconditional as an unexpressed premiss. If the other half of the biconditional is plausibly true, then the argument could be a valid enthymeme.

### Analysis of the Example:

To say that q is a "necessary component" of p is to mean that if one has p one must also have q, that is: "if p then q". For example, "an engine is a necessary component of a functioning automobile" means that if one has a functioning car then one has an engine, rather than if one has an engine then one has a functioning car. So, Morris' argument is as follows:

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

Even if the first premiss were true—which it is not—it doesn't follow from a disbelief in atheism or pantheism that one must disbelieve in evolution. There are many theistic religions which accept evolution as an historical fact.

Notes:

1. A. R. Lacey, A Dictionary of Philosophy (Third Revised Edition) (Barnes & Noble, 1996).
2. Henry M. Morris, The Remarkable Birth of Planet Earth, (Creation-Life Publishers, 1972), pp. vi-vii.