Beth Semantics and Labelled Deduction for Intuitionistic Sentential Calculus with Identity

Authors Didier Galmiche, Marta Gawek, Daniel Méry

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Author Details

Didier Galmiche
  • Université de Lorraine, CNRS, LORIA, Nancy, France
Marta Gawek
  • Université de Lorraine, CNRS, LORIA, Nancy, France
Daniel Méry
  • Université de Lorraine, CNRS, LORIA, Nancy, France

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Didier Galmiche, Marta Gawek, and Daniel Méry. Beth Semantics and Labelled Deduction for Intuitionistic Sentential Calculus with Identity. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper we consider the intuitionistic sentential calculus with Suszko’s identity (ISCI). After recalling the basic concepts of the logic and its associated Hilbert proof system, we introduce a new sound and complete class of models for ISCI which can be viewed as algebraic counterparts (and extensions) of sheaf-theoretic topological models of intuitionistic logic. We use this new class of models, called Beth semantics for ISCI, to derive a first labelled sequent calculus and show its adequacy w.r.t. the standard Hilbert axiomatization of ISCI. This labelled proof system, like all other current proof systems for ISCI that we know of, does not enjoy the subformula property, which is problematic for achieving termination. We therefore introduce a second labelled sequent calculus in which the standard rules for identity are replaced with new special rules and show that this second calculus admits cut-elimination. Finally, using a key regularity property of the forcing relation in Beth models, we show that the eigenvariable condition can be dropped, thus leading to the termination and decidability results.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Algebraic Semantics
  • Beth Models
  • Labelled Deduction
  • Intuitionistic Logic


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