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 :
\begin{array}{c}\Gamma\vdashA,\Delta \Gamma',A\vdash\Delta'\ \hline\Gamma,\Gamma'\vdash\Delta,\Delta'\end{array}
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.