modus tollens

The law of modus tollens is the inference rule which allows one toconclude $\mathrm{\neg }P$ from $P\Rightarrow Q$ and $\mathrm{\neg }Q$. The name “modustollens” refers to the fact that this rule allows one to take away theconclusion of a conditional statement and conclude the negation of thecondition. As an example of this rule, we may cite the following:

$$\frac{\genfrac{}{}{0pt}{}{\text{If the postman is at the door, the doorbell will ring twice}}{\text{The bell is not ringing.}}}{\text{The postman is not at the door.}}$$

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

$P$	$Q$	$P\Rightarrow Q$	$\mathrm{\neg }P$	$\mathrm{\neg }Q$
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 bycontradiction. In this technique, one assumes a hypothesis $P$ andthen derives a conclusion $Q$. This is tantamount to showing that$P\Rightarrow Q$. Next one demonstrates $\mathrm{\neg }Q$. Applying modustollens, one then concludes $\mathrm{\neg }P$.