Modal definability in enriched languages
Ordinary propositional modal logic is studied using structures called frames. A frame is a pair \(<W,R>\) where W is a class and R a relation. An increasingly important area of research is the connection between properties of frames expressed in a predicate language and properties expressed in a modal propositional language. Thus we know that reflexiveness, \(\forall xRxx,\) can be captured by the modal formula \(\square \phi \to \phi\), but that irreflexiveness, \(\forall x-Rxx,\) cannot be expressed by any modal formula. At least that is so unless the modal language is enriched to contain other necessity operators and the frames extended to contain other relations. This paper investigates such extensions, paying special attention to frames of the form \(<W,R,-R>\) in which -R is the complement of R. The modal language contains two necessity operators, and a number of theorems are proved concerning the class of frames which are modally definable. The irreflexiveness of R can now be expressed by the reflexiveness of -R. Adding the converses of R and -R and their associated necessity operators further enlarges the properties of frames expressible in the modal language. Other enlargements are also considered.
scientific article; zbMATH DE number 4158598