Modus tollens
Formal notation

The modus tollens rule can be stated formally as:

where P → Q {\displaystyle P\to Q} stands for the statement "P implies Q". ¬ Q {\displaystyle \neg Q} stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever " P → Q {\displaystyle P\to Q} " and " ¬ Q {\displaystyle \neg Q} " each appear by themselves as a line of a proof, then " ¬ P {\displaystyle \neg P} " can validly be placed on a subsequent line.

The modus tollens rule may be written in sequent notation:

where ⊢ {\displaystyle \vdash } is a metalogical symbol meaning that ¬ P {\displaystyle \neg P} is a syntactic consequence of P → Q {\displaystyle P\to Q} and ¬ Q {\displaystyle \neg Q} in some logical system;

or as the statement of a functional tautology or theorem of propositional logic:

where P {\displaystyle P} and Q {\displaystyle Q} are propositions expressed in some formal system;

or including assumptions:

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:

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

Also in first-order predicate logic:

("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.
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