Modality and Self-Reference

Abstract

Some self-applied protosyntactical systems related to various modal systems are considered in this chapter . Each of these systems contains one predicate variable P , ranging over properties of the expressions of the system. When an interpretation is provided to the symbol P , each sentence becomes true or false. On selecting an arbitrary set of sentences as axioms and an arbitrary set of inference rules, P can be interpreted to mean provability within the very axiom system. This is essentially referred to self-referential interpretation of the axiom system. The chapter considers only those systems in which this self-referential phenomenon occurs—commonly called as “self-referentially correct.” The chapter also describes δ * , which is an analog of the modal system K4 with certain substitution axioms added, which provides enough fixed-points for the arguments of Godel's second-incompleteness theorem and Lob's theorem to go through the full power of the modal system G. Further, the chapter presents a brief description of some self-referential systems related to modal systems other than K4.