modus tollens

The law of modus tollens is the inference rule which allows one to conclude $\mathrm{\neg }P$ from $P\Rightarrow Q$ and $\mathrm{\neg }Q$. 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:

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

The validity of this rule may be established by means of the following truth 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 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 $\mathrm{\neg }Q$. Applying modus tollens, one then concludes $\mathrm{\neg }P$.

Title	modus tollens
Canonical name	ModusTollens
Date of creation	2013-03-22 16:56:03
Last modified on	2013-03-22 16:56:03
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 03B22
Classification	msc 03B35
Classification	msc 03B05