# modus tollens

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:

$P$ $Q$ $P\Rightarrow Q$ $\neg P$ $\neg Q$
F F T T T
F T T T F
T F F F T
T T T F F

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 $\neg P$.

Title modus tollens ModusTollens 2013-03-22 16:56:03 2013-03-22 16:56:03 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 03B22 msc 03B35 msc 03B05