Denying the Antecedent
Brett Gaul
If Sophia is in the Twin Cities, then she is in Minnesota. Sophia is not in the Twin Cities. Therefore, she is not in Minnesota.
Jane Doe
Like affirming the consequent (see Chapter 2), denying the antecedent is also a fallacious form of reasoning in formal logic.
This time the problem occurs when the minor premise of a propositional syllogism denies the antecedent of a conditional statement. Denying the antecedent makes the mistake of assuming that if the antecedent is denied, then the consequent must also be denied. In other words, if the antecedent is not true, then the consequent must not be true either. However, that is not the case. To see this, assume again that p = “Sophia is in the Twin Cities” and that q = “she is in Minnesota.”(1) If Sophia is in the Twin Cities, then she is in Minnesota. (If p, then q)
(2) Sophia is not in the Twin Cities. (not p)
(3) Therefore, Sophia is not in Minnesota. (not q)
If Sophia is not in the Twin Cities, it does not follow that she is not in Minnesota. She could be in many other places in Minnesota.
Just as the invalid affirming the consequent is similar to the valid modus ponens, the invalid denying the antecedent is similar to the valid modus tollens (the mode of taking). While modus tollens denies the consequent of a conditional statement, denying the antecedent denies the antecedent of a conditional statement.
| Modus tollens (valid) | Denying the antecedent (invalid) |
| If p, then q. | If p, then q. |
| Not q. | Not p. |
| Therefore, not p. | Therefore, not q. |
Like modus ponens, modus tollens is a valid argument form because the truth of the premises guarantees the truth of the conclusion; however, like affirming the consequent, denying the antecedent is an invalid argument form because the truth of the premises does not guarantee the truth of the conclusion. Given the form of modus tollens, if Sophia is not in Minnesota (not q), then she is not in the Twin Cities (not p), because the Twin Cities are in Minnesota.
It’s impossible for her to be in the Twin Cities if she is not in Minnesota, because being in Minnesota is a necessary condition - one that is needed or required - for being in the Twin Cities. However, being in Minnesota is not a sufficient condition - one that guarantees a result - for being in the Twin Cities, because being in Minnesota does not guarantee that one is in the Twin Cities. Nor is being in the Twin Cities necessary for being in Minnesota.The invalidity of denying the antecedent is confirmed by a truth table.
| Premise | Premise | Conclusion |
| If p, then q | Not p | Not q |
| True | False | False |
| False | False | True |
| True | True | False |
| True | True | True |
The third row of the truth table indicates that the argument form allows true premises (If p, then q and not p) and a false conclusion (not q). For an argument to be valid, though, it has to be impossible for the premises to be true and the conclusion to be false. Thus, denying the antecedent is an invalid argument form.