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In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. Its meaning is that, if a formula A appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula A does not appear can be deduced. This applies to cases of modus ponens, such as how instances of man are eliminated from Every man is mortal, Socrates is a man to deduce Socrates is mortal.

It is normally written in formal notation in sequent calculus notation as :

${\displaystyle {\begin{array}{c}\Gamma \vdash A,\Delta \quad \Gamma ',A\vdash \Delta '\\\hline \Gamma ,\Gamma '\vdash \Delta ,\Delta '\end{array}}}$cut^[1]

The cut rule is the subject of an important theorem, the cut-elimination theorem. It states that any sequent that has a proof in the sequent calculus making use of the cut rule also has a cut-free proof, that is, a proof that does not make use of the cut rule.

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