Implications-as-Rules vs. Implications-as-Links: An Alternative Implication-Left Schema for the Sequent Calculus

The interpretation of implications as rules motivates a different left-introduction schema for implication in the sequent calculus, which is conceptually more basic than the implication-left schema proposed by Gentzen. Corresponding to results obtained for systems with higher-level rules, it enjoys the subformula property and cut elimination in a weak form. The introduction schema for implication on the left side of the turnstile in Gentzen’s sequent calculus for intuitionistic logic runs as follows: (→L) Γ⊢A ∆, B ⊢C Γ,∆, A→B ⊢C . With all other logical inference schemata it shares (among others) the following two properties: (I) The schema contains exactly one connective. (II) The connective occurs only in the conclusion of the schema, i.e. below the inference line, and there exactly once. ∗This paper was written during a research stay at IHPST Paris supported by the Fondation Maison des Sciences de l’Homme, by the ESF research project “Dialogical Foundations of Semantics (DiFoS)” within the ESF-EUROCORES programme “LogICCC — Modelling Intelligent Interaction” (DFG Schr 275/15-1) and the French-German DFG-ANR project “Hypothetical Reasoning — Logical and Semantical Perspectives (HYPOTHESES)” (DFG Schr 275/16-1). I gratefully acknowledge the comments of participants of a seminar given at IHPST on the subject of this paper.