Information Logics Versus Standard Modal Logics
In this chapter we investigate relationships between the Rare-logics introduced in Sect. 5.4 and standard modal logics defined in Sect. 5.3.3. For the sake of simplicity, we confine ourselves to Rare-logics with a unique relation type, that is, the models of the logics contain a single family of relative rela tions. The focus is on studying the conditions which enable us to reduce the level of the powerset hierarchy of parameters in the semantic structures of the logics without changing the set of valid formulae. In Sect. 10.2 we introduce several classes of Rare-logics by postulating that the relations from semantic structures satisfy various global conditions, in particular the conditions listed in Sect. 3.9. In Sect. 10.3 and in Sect. 10.4 we show how formulae of some Rare-logics can be translated into formulae of certain standard modal logics with preservation of validity. The standard modal logics associated in this way with Rare-logics must be sufficiently expressive; in particular, often they must include the universal modal connective. In Sect. 10.5 we address the issue of whether the universal modal connective can be eliminated from the language without violating the properties of the translation. Sect. 10.6 and Sect. 10.7 extend the results and techniques from Sect. 10.3 and Sect. 10.4 to the Rare-logics whose languages include constants representing individual objects and individual parameters.
KeywordsNormal Form Global Condition Propositional Variable Modal Expression Semantic Structure
Unable to display preview. Download preview PDF.