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Cut-free labelled calculi and decidability for intuitionistic sentential logic with identity.

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Title: Cut-free labelled calculi and decidability for intuitionistic sentential logic with identity.
Authors: Galmiche, Didier1 (AUTHOR), Hornbeck, Brandon1 (AUTHOR), Méry, Daniel1 (AUTHOR)
Source: Journal of Logic & Computation. Jun2025, Vol. 35 Issue 4, p1-51. 51p.
Subjects: Propositional calculus, Calculi, Logic, Semantics, Automation
Abstract: In this paper we consider the intuitionistic non-Fregean sentential calculus with Suszko's identity (⁠|$\mathsf{ISCI}$|⁠). 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 |$\mathsf{ISCI}$|⁠ , called Topological Beth (⁠|${\mathsf{TB}}$|⁠) models, that can be viewed as algebraic counterparts (and extensions) of sheaf-theoretic topological models of intuitionistic logic. From this semantical study we define a family of sound and cut-free complete labelled calculi that capture both the Kripke and the |${\mathsf{TB}}$| semantics. Using a key property of the forcing relation in |${\mathsf{TB}}$| models, called regularity, we show termination and decidability results. Finally we discuss the automation of the proof search in one of the labelled calculi and its implementation. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Cut-free labelled calculi and decidability for intuitionistic sentential logic with identity.
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  Data: In this paper we consider the intuitionistic non-Fregean sentential calculus with Suszko's identity (⁠|$\mathsf{ISCI}$|⁠). 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 |$\mathsf{ISCI}$|⁠ , called Topological Beth (⁠|${\mathsf{TB}}$|⁠) models, that can be viewed as algebraic counterparts (and extensions) of sheaf-theoretic topological models of intuitionistic logic. From this semantical study we define a family of sound and cut-free complete labelled calculi that capture both the Kripke and the |${\mathsf{TB}}$| semantics. Using a key property of the forcing relation in |${\mathsf{TB}}$| models, called regularity, we show termination and decidability results. Finally we discuss the automation of the proof search in one of the labelled calculi and its implementation. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1093/logcom/exae071
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        Text: English
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        PageCount: 51
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      – SubjectFull: Propositional calculus
        Type: general
      – SubjectFull: Calculi
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      – SubjectFull: Logic
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      – TitleFull: Cut-free labelled calculi and decidability for intuitionistic sentential logic with identity.
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              M: 06
              Text: Jun2025
              Type: published
              Y: 2025
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