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Modal logics with Belnapian truth values

机译：具有Belnapian真值的模态逻辑

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摘要

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of first-degree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification (support of truth) and the other for falsification (support of falsity). Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation.
机译：研究了各种四值和三值模态命题逻辑。基本系统是Belnap的模态扩展BK和BS4，以及Dunn的一级蕴涵的四值逻辑。还考虑了BK和BS4的三值扩展。这些逻辑是通过具有两个不同评估关系的关系模型在语义上引入的，一个用于验证（支持事实），另一个用于伪造（支持虚假）。公理系统相对于关系语义以及模态代数上的扭曲结构，被定义并显示为健全且完整的。声音和完整的表格结石也被介绍。此外，许多具有强否定性的构造性非模态逻辑被忠实地嵌入到BS4，其三值扩展B3S4或时间BS4，BtS4中。这些逻辑包括戴维·纳尔逊（David Nelson）的三值逻辑N3，四值逻辑N4，连接逻辑C，以及通过强取反得到的双直觉逻辑的若干扩展。

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