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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. 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 in sequent calculus notation :
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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In propositional logic, modus ponens, also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore, Q must also be true."
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The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus making use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule.
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide decision procedures, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to model theory.
In logic, a rule of inference is admissible in a formal system if the set of theorems of the system does not change when that rule is added to the existing rules of the system. In other words, every formula that can be derived using that rule is already derivable without that rule, so, in a sense, it is redundant. The concept of an admissible rule was introduced by Paul Lorenzen (1955).
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