We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

中文翻译：

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BELNAP-DUNN 模态逻辑：真值常数 VS。真实价值观

我们将关注模态逻辑 BK——它基于 Belnap-Dunn 四值矩阵，可以看作是从最小正态模态逻辑 K 中加入“强否定”得到的。尽管在其 Kripke 语义中使用了所有四个值“真”、“假”、“两者”和“两者”，但只有前两个可以表示为术语。我们表明，扩展 BK 的原始语言以包含“既不”或/和“两者”的常量会导致非常意想不到的结果。更准确地说，添加这些常数之一具有消除 BK 扩展级别的相应值的效果。特别是，如果将这两者相加，则相应的扩展格与普通普通模态逻辑的扩展格是同构的。