Abstract
This chapter introduces the original sequent calculus of Gentzen, called LK (der Logistische Kalkül). In sections 2.2–2.3. we investigate some applications of cut concerning the equivalence of some forms of sequents and sequent rules and some invertibility results. Then we focus on different strategies of proving the cut elimination/admissibility theorem, dividing them on local and global proofs. The former are based on small transformations of particular steps of a proof. The latter are based on transformations of the whole proofs of the premisses of cut. Section 2.6 and 2.7 contain a discussion of decidability proof for LK and Kleene’s result concerning permutability of rules.
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Several proposals of this sort were offered for different classes of SC. For example by Avron [14] or Ciabattoni [50], where the notion of substitutive and reductive rules was introduced. Ciabattoni and Terui [51] provide a survey of such attempts.
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We treat both variants of \((\wedge \Rightarrow )\) and \((\Rightarrow \vee )\) as one rule.
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Indrzejczak, A. (2021). Gentzen’s Sequent Calculus LK. In: Sequents and Trees. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-57145-0_2
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DOI: https://doi.org/10.1007/978-3-030-57145-0_2
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