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This paper considers the bi-modal logic with both \(\Box \)  and \(\Diamond \) arising from Kripke models with crisp accessibility whose propositions are valued over the standard Gödel algebra [0, 1]. Since this logic lacks the finite model property, we study the logic \(\textbf{GW}^\textrm{c}\) relying on witnessed Kripke models where, for each modal formula, there is an assignment where the formula without the modality takes the same value as the modal one. We provide a cut-free sequent calculus and we exploit it to prove that \(\textbf{GW}^\textrm{c}\) is decidable and meets the finite model property. Finally, we explore a connection between the witnessed models and the well-known bi-relational Kripke semantics.

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A Gödel Modal Logic over Witnessed Crisp Models

  • 会议论文

  • 出版模式：

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  • 发表日期：

    2026年

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  • Mauro Ferrari,
  • Camillo Fiorentini,
  • Ricardo Oscar Rodriguez

摘要

This paper considers the bi-modal logic with both \(\Box \)  and \(\Diamond \) arising from Kripke models with crisp accessibility whose propositions are valued over the standard Gödel algebra [0, 1]. Since this logic lacks the finite model property, we study the logic \(\textbf{GW}^\textrm{c}\) relying on witnessed Kripke models where, for each modal formula, there is an assignment where the formula without the modality takes the same value as the modal one. We provide a cut-free sequent calculus and we exploit it to prove that \(\textbf{GW}^\textrm{c}\) is decidable and meets the finite model property. Finally, we explore a connection between the witnessed models and the well-known bi-relational Kripke semantics.