We present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown.

中文翻译：

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各种基于 FDE 的模态逻辑的证明系统

我们为各种基于 FDE 的模态逻辑提出了新颖的证明系统。所考虑的系统包括在 Odintsov & Wansing (2010) 和 Odintsov & Wansing (2017) 中引入的许多 Belnapian 模态逻辑，以及在 Goble (2006) 中引入的具有强烈含义的模态逻辑 KN4。特别是，我们为逻辑提供了一个希尔伯特式公理系统$BK^{\square - } $并将逻辑 BK 描述为系统的公理扩展$BK^{FS} $. 对于 KN4，我们提供了 FDE 式公理系统和可判定的连续演算，其中显示了收缩消除和切割消除结果。