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In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/; MT; also modus tollendo tollens (Latin for "mode that denies by denying")[1] or denying the consequent)[2] 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 inference rule modus tollens validates the inference from
P
P implies
Q
Q and the contradictory of
Q
Q to the contradictory of
P
P.
The modus tollens rule can be stated formally as:
P
→
Q
,
¬
Q
∴
¬
P
{\frac {P\to Q,\neg Q}{\therefore \neg P}}
where
P
→
Q
P\to Q stands for the statement "P implies Q".
¬
Q
\neg Q stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "
P
→
Q
P\to Q" and "
¬
Q
\neg Q" each appear by themselves as a line of a proof, then "
¬
P
\neg P" can validly be placed on a subsequent line. The history of the inference rule modus tollens goes back to antiquity.[3]
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.
The first to explicitly describe the argument form modus tollens was Theophrastus.[4]
<<There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent
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Modus tollens