Formal Notation
The modus tollens rule may be written in sequent notation:
where is a metalogical symbol meaning that is a syntactic consequence of and in some logical system;
or as the statement of a functional tautology or theorem of propositional logic:
where, and are propositions expressed in some logical 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. There exists some x that is not Q. Therefore, there exists some x that is not P.")
Strictly speaking these are not instances of modus tollens, but they may be derived using modus tollens using a few extra steps.
Read more about this topic: Modus Tollens
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