How does denying the antecedent produce an invalid conclusion?

Denying the antecedent is a formal fallacy that produces an invalid conclusion because it incorrectly assumes that the truth of a consequent depends solely on its antecedent.

In a conditional statement, such as "If $p_1$, then $p_2$," the antecedent ($p_1$) is a sufficient condition for the consequent ($p_2$), but it is not necessarily a necessary one. Denying the antecedent involves arguing from the premises "If $p_1$, then $p_2$" and "not-$p_1$" to the conclusion "not-$p_2$."1

This reasoning is invalid because the consequent ($p_2$) could potentially be true for reasons other than the occurrence of the antecedent ($p_1$). For example, in the statement "If it rains, the game is canceled," the game might still be canceled for other reasons (such as a power outage or a scheduling conflict) even if it does not rain. Therefore, negating the antecedent does not logically necessitate the negation of the consequent. This error stands in contrast to modus tollens, a valid form of reasoning where the consequent is denied to correctly conclude that the antecedent must also be false.23