Cut Rule

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:
: $\begin\Gamma \vdash A, \Delta \\ \Gamma', A \vdash \Delta' \\ \hline \Gamma, \Gamma' \vdash \Delta, \Delta'\end{array}$

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.

Rules of inference
Logical calculi