Modus tollens

In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.

Modus tollens

Type	Deductive argument form Rule of inference
Field	Classical logic Propositional calculus
Statement	${\displaystyle P}$ implies ${\displaystyle Q}$. ${\displaystyle Q}$ is false. Therefore, ${\displaystyle P}$ must also be false.
Symbolic statement	${\displaystyle P\rightarrow Q,\neg Q}$ ${\displaystyle \therefore \neg P}$

The history of the inference rule modus tollens goes back to antiquity. The first to explicitly describe the argument form modus tollens was Theophrastus.

Modus tollens is closely related to modus ponens. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent. See also contraposition and proof by contrapositive.