We investigate some non-normal variants of well-studied paraconsistent and paracomplete modal logics that are based on N. Belnap’s and M. Dunn’s four-valued logic. Our basic non-normal modal logics are characterized by a weak extensionality rule, which reflects the four-valued nature of underlying logics. Aside from introducing our basic framework of bi-neighbourhood semantics, we develop a correspondence theory in order to prove completeness results with respect to our neighbourhood semantics for non-normal variants of \(\mathsf {BK}\), \(\mathsf {BK^{FS}}\) and \(\mathsf {MBL}\).

中文翻译：

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基于FDE的模态逻辑的邻域语义

我们研究了基于N. Belnap和M. Dunn的四值逻辑的，经过充分研究的超常和超完全模态逻辑的一些非正规变体。我们的基本非正规模态逻辑的特征在于弱的可扩展性规则，该规则反映了基础逻辑的四值性质。除了介绍我们的双邻域语义的基本框架之外，我们还开发了一种对应论，以证明\（\ mathsf {BK} \），\（\ mathsf { BK ^ {FS}} \）和\（\ mathsf {MBL} \）。