FDE-based modal logics

该项目旨在发展和研究某些非经典逻辑系统，这些系统在一些基本性质方面与经典逻辑不同。非经典逻辑的研究是由来自数学基础、人工智能、自然语言语义和解决哲学逻辑中分析的悖论等领域的需求所驱动的。更具体地说，该项目致力于研究扩展一级蕴涵逻辑（FDE）的模态逻辑，FDE是一种基本的形式系统，也被称为Belnap-Dunn逻辑。系统FDE是一个非常突出的非经典逻辑。它是一种多值和超一致的关联逻辑系统，具有许多应用，例如在计算机科学中。模态运算符对FDE的扩展是特别有趣的，因为这些运算符带有各种读数，例如“有必要”，“有可能”，“已知”，“有义务”等。命题FDE以某些四值真值表为特征，这些真值表利用了语义值，这些语义值是根据各种来源提供的关于原子陈述的语义状态的信息来最好地理解的：一个陈述可能只被告知为真，它可能只被告知为假，它可能既不被告知为真也不被告知假，或者它可能既被告知为真又被告知为假。介绍和研究了基本系统FDE的各种模态和非模态扩展。一个特别重要的是O. Arieli和A. Avron的逻辑双边逻辑。本课题的出发点是模态逻辑BK，它可以表示为最小正态模态命题逻辑K的保守扩展，也可以表示为命题FDE的扩展。FDE的其他模态扩展已经被研究过，包括一个叫做BN4的系统，以及最近由a . Jung和U. Rivieccio引入的模态双格逻辑MBL。在后一种系统中，由于信息状态之间的可访问性关系也是四值的，因此语义更加激进地多值化。最近，S.P. Odintsov和H. Wansing对中央系统BK、BN4和MBL之间的形式关系进行了研究，并在很大程度上澄清了这些关系。事实证明，定义等价的概念在比较这些逻辑时起着重要的作用，而且由于自拓性的失效，这个概念需要进行一些调整。这一特点提出了许多哲学和数学问题，这些问题将在波鸿和新西伯利亚的两个研究团队之间密切合作，结合哲学和数学逻辑方面的专业知识来解决。该项目的目标包括调查上述逻辑的新证明系统，以及BK， BN4和MBL附近系统的证明理论和代数研究，包括所谓的非正态模态逻辑。

The project aims at the development and investigation of certain nonclassical logical systems that differ from classical logic with respect to a number of fundamental properties. The study of nonclassical logics is motivated by demands coming from areas such as the foundations of mathematics, artificial intelligence, natural language semantics, and resolving paradoxes analyzed in philosophical logic. More specifically, the project pursues the investigation of modal logics that extend first-degree entailment logic, FDE, a basic formal system also known as Belnap-Dunn logic. The system FDE is a very prominent non-classical logic. It is a many-valued and paraconsistent system of relevance logic that has numerous applications, for example in computer science. Extensions of FDE by modal operators are of special interest because these operators come with various readings such as "it is necessary that", "it is possible that", "it is known that", "it is obligatory that" etc. Propositional FDE is characterized by certain four-valued truth tables which make use of semantical values that are best understood in terms of information provided by various sources concerning the semantic status of atomic statements: a statement may be told only to be true, it may be told only to be false, it may neither be told true nor false, or it may happen that the statement is both told true and told false. Various modal and non-modal extensions of the basic system FDE have been introduced and investigated. A particularly important one is O. Arieli and A. Avron's logic of logical bilattices. The starting point of the present project is the modal logic BK. It can be presented as a conservative extension of the smallest normal modal propositional logic K, but also as an extension of propositional FDE. Other modal extensions of FDE have been investigated, including a system called BN4 and, more recently, the modal bilattice logic MBL introduced by A. Jung and U. Rivieccio. In the latter system the semantics is more radically many-valued insofar as the accessibility relation between information states is four-valued as well. Recently, the formal relationships between the central systems BK, BN4 and MBL have been investigated and to a large extent clarified by S.P. Odintsov and H. Wansing. It turned out that the notion of definitional equivalence plays an important role for comparing these logics and that this notion calls for some adjustments due to the failure of a property called self-extensionality. This feature raises a number of philosophical and mathematical problems that will be tackled in close collaboration between two research teams in Bochum and Novosibirsk, combining expertise in philosophical and mathematical logic. The objectives of the project include the investigation of novel proof systems for the mentioned logics as well as proof-theoretic and algebraic studies of systems in the vicinity of BK, BN4 and MBL, including so-called non-normal modal logics.