Editing Cut rule

{{Short description|Inference rule}}
{{onesource|date=July 2017}}
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''.

== Formal notation ==
It is normally written in formal notation in sequent calculus notation as :
:
: <math>
\begin{array}{c}\Gamma \vdash A, \Delta \quad  \Gamma', A \vdash \Delta' \\ \hline \Gamma, \Gamma' \vdash \Delta, \Delta'\end{array} </math>cut<ref>{{Cite web |title=cut rule in nLab |url=https://ncatlab.org/nlab/show/cut+rule |access-date=2024-10-22 |website=ncatlab.org |language=en}}</ref>

== Elimination ==
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.

==References==
{{reflist}}

[[Category:Rules of inference]]
[[Category:Logical calculi]]


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