A Three-Valued Approach to Overcome Paradoxes in Syntactic Theory

A powerful syntactic theory as well as expressive modal logics have to deal with self-referentiality. Self-referentiality and paradoxes seem to be close neighbours and depending on the logical system, they have devastating consequences, since they introduce contradictions and trivialise the logical system. There is a large amount of diierent attempts to tackle these problems. Some of them are compared in this paper, futhermore a simple approach based on a three-valued logic is advocated. In this approach paradoxes may occur and are treated formally. However, it is necessary to be very careful, otherwise a system built on such an attempt trivialises as well. In order to be able to formally deal with such a system, the reason for self-referential paradoxes is studied in more detail and a semantical condition on the connectives is given such that paradoxes are excluded.