Generalized Elimination Inferences, Higher-Level Rules, and the Implications-as-Rules Interpretation of the Sequent Calculus

  • Peter Schroeder-HeisterEmail author
Part of the Trends in Logic book series (TREN, volume 39)


We investigate the significance of higher-level generalized elimination rules as proposed by the author and compare them with standard-level generalized elimination rules as proposed by Dyckhoff, Tennant, López-Escobar and von Plato. Many of the results established for natural deduction with higher-level rules such as normalization and the subformula principle immediately translate to the standard-level case. The sequent-style interpretation of higher-level natural deduction as proposed by Avron and by the author leads to a system with a weak rule of cut, which enjoys the subformula property. The interpretation of implications-as-rules motivates a different left-introduction schema for implication in the ordinary (standard-level) sequent calculus, which conceptually is more basic than the implication-left schema proposed by Gentzen. Corresponding to the result for the higher-level system, it enjoys the subformula property and cut elimination in a weak form.


Sequent Calculus High-level Rules Generalized Elimination Rules Natural Deduction Gentzen 
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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

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