What is "modus tollens"

n. (context philosophy logic English) A valid form of argument in which the consequent of a conditional proposition is deny, thus implying the denial of the antecedent. Modus tollens has this form:

In propositional logic, modus tollens (or modus tollendo tollens and also denying the consequent) ( Latin for "the way that denies by denying") is a valid argument form and a rule of inference. It is an application of the general truth that if a statement is true, then so is its contra-positive.

The first to explicitly describe the argument form modus tollens were the Stoics.

The inference rule modus tollens validates the inference from P implies Q and the contradictory of Q to the contradictory of P.

The modus tollens rule can be stated formally as:

$$\frac{P \to Q, \neg Q}{\therefore \neg P}$$

where P → Q stands for the statement "P implies Q". ¬Q stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "P → Q" and "¬Q" each appear by themselves as a line of a proof, then "¬P" can validly be placed on a subsequent line. The history of the inference rule modus tollens goes back to antiquity.

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.