Modal Incompleteness Revisited

摘要

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some Σ $ subseteq $ L and some $varphi notin L,varphi $ cannot be separated from Σ by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.