Model Theory for Graded Modal Logic
In the first place, I give an introduction to graded modal logic (GML), including the motivation, a short history of GML and some applications of graded modalities.Later we introduce two ontologies for the semantics of GML: one is the ontology of Kripke frames, and the other is the ontology of Omega-coalgebras which is defined on the category of sets plus the Omega-functor.
Under Kripke semantics, there are already several results in model theory for GML, like frame definability theorem, van Benthem style characterization theorem, and Lindstrom characterization theorem. I will show a theorem on definability of Kripke models in GML, which is a generalization of Yde Venema's theorem for basic modal logic.
Under coalgebraic semantics for GML, I give some constructions of coalgebras and some basics of graded modal algebras. Then by duality between coalgebra and graded modal algebra, I show a Goldblatt-Thomason theorem for GML, i.e., a class of coalgebras which is closed under Omega-ultrafilter extension is definable in GML iff it is closed under sums, generated subcoalgebras and Omega-homomorphic images, while its complement is closed under Omega-ultrafilter extensions.
Next I show completeness theory and correspondence theory for coalgebraic GML. For the completeness theory, I show that the minimal graded modal logic K_g is strongly complete with respect to the class of all Omega-coalgebras. Then I give some normal extensions of K_g. For the correspondence theory, I show that coalgebraic GML corresponds to weak second-order logic, i.e., the second--order logic in which all variables range over the non-empty finite subsets of the domain.
Finally, I give some conclusions and further directions in model theory for GML.
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