Editing Modus tollens (section)

== Formal notation ==
The ''modus tollens'' rule can be stated formally as:

:<math>\frac{P \to Q, \neg Q}{\therefore \neg P}</math>

where <math>P \to Q</math> stands for the statement "P implies Q". <math>\neg  Q</math> stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "<math>P \to Q</math>" and "<math>\neg Q</math>" each appear by themselves as a line of a [[formal proof|proof]], then "<math>\neg P</math>" can validly be placed on a subsequent line.

The ''modus tollens'' rule may be written in [[sequent]] notation:

:<math>P\to Q, \neg Q \vdash \neg P</math>

where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>\neg P</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math> and <math>\neg Q</math> in some [[formal system|logical system]];

or as the statement of a functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic:

:<math>((P \to Q) \land \neg Q) \to \neg P</math>

where <math>P</math> and <math>Q</math> are propositions expressed in some [[formal system]];

or including assumptions:

:<math>\frac{\Gamma \vdash P\to Q ~~~ \Gamma \vdash \neg Q}{\Gamma \vdash \neg P}</math>

though since the rule does not change the set of assumptions, this is not strictly necessary.

More complex rewritings involving ''modus tollens'' are often seen, for instance in [[set theory]]:

:<math>P\subseteq Q</math>
:<math>x\notin Q</math>
:<math>\therefore x\notin P</math>

("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

Also in first-order [[predicate logic]]:

:<math>\forall x:~P(x) \to Q(x)</math>
:<math>\neg Q(y)</math>
:<math>\therefore ~\neg P(y)</math>

("For all x if x is P then x is Q. y is not Q. Therefore, y is not P.")

Strictly speaking these are not instances of ''modus tollens'', but they may be derived from ''modus tollens'' using a few extra steps.