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Terminating Calculi and Countermodels for Constructive Modal Logics

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)


We investigate terminating sequent calculi for constructive modal logics \(\mathsf {CK}\) and \(\mathsf {CCDL}\) in the style of Dyckhoff’s calculi for intuitionistic logic. We first present strictly terminating calculi for these logics. Our calculi provide immediately a decision procedure for the respective logics and have good proof-theoretical properties, namely they allow for a syntactic proof of cut admissibility. We then present refutation calculi for non-provability in both logics. Their main feature is that they support direct countermodel extraction: each refutation directly defines a finite countermodel of the refuted formula in a natural neighbourhood semantics for these logics.

We thank the reviewers for very accurate comments and corrections that helped us to improve the first version of this paper. This work has been partially supported by the ANR-FWF project TICAMORE ANR-16-CE91-0002-01; FWF I 2982. Dalmonte is supported by a Ernst Mach worldwide grant implemented by the OeAD, Austria Agency for Education and Internationalisation, and financed by BMBWF.

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  1. 1.

    Decidability for these logics follows from the finite model property established in Mendler and de Paiva [16] and Dalmonte et al. [3].

  2. 2.

    To be precise, the sets \(\varGamma ^\sigma \) and \(\varDelta ^\sigma \) depend on the refutation \(\mathscr {R}\). In order not to burden the notation we avoid explicit reference to \(\mathscr {R}\) as it is clear from the context.

  3. 3.

    The transformation in [3] must be slightly modified given the alternative formulation of the neighbourhood semantics.

  4. 4.

    As an example, an extraction of relational countermodels from failed proofs in a G4-calculus for Intuitionistic Strong Löb Logic with only \(\Box \) is presented in [9].


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Dalmonte, T., Grellois, C., Olivetti, N. (2021). Terminating Calculi and Countermodels for Constructive Modal Logics. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham.

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