FDE-based modal logics

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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.