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. In the particular case of the modus ponens, for example occurrences of man are eliminated of Every man is mortal, Socrates is a man to deduce Socrates is mortal.

Formal notation

Formal notation in sequent calculus notation :

cut

${\displaystyle {\begin{array}{l}\Gamma \vdash A,\Delta \\\Gamma ',A\vdash \Delta '\\\hline \Gamma ,\Gamma '\vdash \Delta ,\Delta '\end{array))}$

Elimination

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