Filtration Revisited: Lattices of Stable Non-Classical Logics
Julia Ilin
Abstract:
The topics covered in the thesis are about intuitionistic and modal logics but also touch the area of dynamic epistemic logic.
There are two standard methods to prove the finite model property (fmp) in modal and intuitionisitic logics: one of them is filtration and another one is selective filtration. Selective filtration leads to the notion of subframes, subframe formulas and subframe logics. Subframe logics are a well-studied class of modal and superintuitionistic logics.
In this thesis, we concentrate on the notion of filtration and classes of formulas and logics that it gives rise to. Such logics are called stable logics. We study properties of stable logics and compare them to those of subframe logics. We also study subframe and stable intermediate logics via translations into intuitionistic modal logic. We apply tools from algebra and duality theory.
Another chapter is devoted to the class of NNIL-formulas. NNIL-formulas are defined by their syntactic shape (no nesting of implication to the left) and are known to axiomatize all intuitionistic subframe logics. Among our central results here is the construction of a universal model for NNIL-formulas.
Finally, we study the incarnation of filtration/stability in the setting of dynamic epistemic logic. We investigating a modality whose corresponding model transformation corresponds to quotienting. Epistemically, quotienting can be thought of as an abstraction. We investigate technical properties of logics equipped with such abstraction modalities. In some special cases, such logics can be regarded as logics of filtrations.