A Note on the Model Theory for Positive Modal Logic

The minimum system of Positive Modal Logic SK+ is the (∧, ∨, □, ◊, ⊥, $\top$)-fragment of the minimum normal modal logic K with local consequence. In this paper we develop some of the model theory for SK+ along the yet standard lines of the model theory for classical normal modal logic. We define the notion of positive bisimulation between two models, and we study the notions of m-saturated models and replete models. We investigate the positive maximal Hennessy-Milner classes. Finally, we present a Keisler-Shelah type theorem for positive bisimulations, a characterization of the first-order formulas invariant for positive bisimulations, and two definability theorems by positive modal sequents for classes of pointed models.