The law of modus tollens is the inference rule which allows one to conclude $\neg P$ from $P \Rightarrow Q$ and $\neg Q$ . The name ``modus tollens'' refers to the fact that this rule allows one to take away the conclusion of a conditional statement and conclude the negation of the condition. As an example of this rule, we may cite the following:$$ {{\hbox{If the postman is at the door, the doorbell will ring twice} \atop \hbox{The bell is not ringing.}} \over \hbox{The postman is not at the door.}}$$

The validity of this rule may be established by means of the following truth table:

This rule can be used to justify the popular technique of proof by contradiction. In this technique, one assumes a hypothesis $P$ and then derives a conclusion $Q$ . This is tantamount to showing that $P \Rightarrow Q$ . Next one demonstrates $\neg Q$ . Applying modus tollens, one then concludes $\neg P$ .