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. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 0955792X |
| DOI: | 10.1093/logcom/exae071 |
