The language of Belnap–Dunn modal logic $${\mathscr {L}}_0$$ expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator $$\Box $$ . We introduce the polarity semantics for $${\mathscr {L}}_0$$ and its two expansions $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ with value operators. The local finitary consequence relation $$\models _4^k$$ in the language $${\mathscr {L}}_k$$ with respect to the class of all frames is axiomatized by a sequent system $$\mathsf {S}_k$$ where $$k=0, 1, 2$$ . We prove by using translations between sequents and formulas that these languages under the polarity semantics have the same expressive power on the level of frames with the language $${\mathscr {L}}_0$$ under the relational semantics for classical modal logic.

中文翻译：

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带值运算符的 Belnap-Dunn 模态逻辑

Belnap–Dunn 模态逻辑语言 $${\mathscr {L}}_0$$ 使用模态运算符 $$\Box 扩展了 Belnap–Dunn 四值逻辑（具有值 0 和 1 的常量符号）的语言$$ 。我们介绍了 $${\mathscr {L}}_0$$ 及其两个扩展 $${\mathscr {L}}_1$$ 和 $${\mathscr {L}}_2$$ 的极性语义，以及值运算符. 语言 $${\mathscr {L}}_k$$ 中关于所有帧的类的局部有限结果关系 $$\models _4^k$$ 由序列系统 $$\mathsf {S} 公理化_k$$ 其中 $$k=0, 1, 2$$ 。我们通过使用序列和公式之间的翻译证明，极性语义下的这些语言在框架级别上与经典模态逻辑的关系语义下的语言 $${\mathscr {L}}_0$$ 具有相同的表达能力。