Modus tollens

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	${\displaystyle P\rightarrow Q,\neg Q}$ ${\displaystyle \therefore \neg P}$[1]

In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "method of removing by taking away")^[2] and denying the consequent,^[3] 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.

The history of the inference rule modus tollens goes back to antiquity.^[4] The first to explicitly describe the argument form modus tollens was Theophrastus.^[5]

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.