In
propositional logic,
modus tollens[1][2][3][4] (or
modus tollendo tollens and also
denying the consequent)
[5] (
Latin for "the way that denies by denying")
[6] is a
validargument 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.
[7]
The inference rule
modus tollens validates the
inference from {\displaystyle P}
implies {\displaystyle Q}
and the contradictory of {\displaystyle Q}
to the contradictory of {\displaystyle P}
.
The
modus tollens rule can be stated formally as:
{\displaystyle {\frac {P\to Q,\neg Q}{\therefore \neg P}}}
where {\displaystyle P\to Q}
stands for the statement "P implies Q". {\displaystyle \neg Q}
stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "{\displaystyle P\to Q}
" and "{\displaystyle \neg Q}
" each appear by themselves as a line of a
proof, then "{\displaystyle \neg P}
" can validly be placed on a subsequent line. The history of the inference rule
modus tollens goes back to antiquity.
[8]
Modus tollens is closely related to
modus ponens. There are two similar, but
invalid, forms of argument:
affirming the consequent and
denying the antecedent.