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FDE-based modal logics

基于 FDE 的模态逻辑

基本信息

批准号：
389151720
负责人：
Professor Dr. Heinrich Wansing
金额：
--
依托单位：
Sobolev Institute of Mathematics
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/389151720?language=en
关键词：
FDE based modal logics

项目摘要

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.

该项目旨在开发和研究某些非经典逻辑系统，这些系统在一些基本属性方面不同于经典逻辑。非经典逻辑的研究是由来自数学基础，人工智能，自然语言语义学和解决哲学逻辑中分析的悖论等领域的需求推动的。更具体地说，该项目追求模态逻辑的研究，扩展了一级蕴涵逻辑，FDE，一个基本的形式系统，也被称为Belnap-Dunn逻辑。系统FDE是一个非常突出的非经典逻辑。它是一个多值和次协调的关联逻辑系统，有许多应用，例如在计算机科学中。通过模态操作符对FDE进行扩展是特别有趣的，因为这些操作符带有各种读数，例如“必须”，"可能”，“已知”，“这是强制性的，”等等。命题FDE的特点是某些四个-值真值表，其利用语义值，这些语义值根据由各种源提供的关于语义状态的信息而被最好地理解原子语句：一个陈述可以只被告知是真的，也可以只被告知是假的，既可以不被告知是真的也可以不被告知是假的，或者可能发生的是，这个陈述既被告知是真的又被告知是假的。各种模态和非模态扩展的基本系统FDE已被介绍和研究。一个特别重要的是O。Arieli和A.逻辑双格的逻辑。本项目的出发点是模态逻辑BK。它可以被表示为最小正规模态命题逻辑K的保守扩展，也可以被表示为命题FDE的扩展。FDE的其他模态扩展也被研究，包括一个称为BN 4的系统，以及最近由A. Jung和U.里维乔在后一种系统中，语义是更根本的多值的，因为信息状态之间的可访问性关系也是四值的。最近，S. P. Odintsov和H.万辛事实证明，定义等价的概念在比较这些逻辑中起着重要的作用，并且由于一个称为自延性的属性的失败，这个概念需要一些调整。这一特点提出了一些哲学和数学问题，将在波鸿和新西伯利亚的两个研究小组密切合作，结合哲学和数学逻辑的专业知识。该项目的目标包括研究上述逻辑的新证明系统，以及BK，BN 4和MBL附近系统的证明理论和代数研究，包括所谓的非正规模态逻辑。