Part I. Representation of Boolean algebras as fields of sets. Same for distributive lattices. Logical interpretation of the representation theorems (soundness and completeness). Topological interpretation of the representation theorems (Stone duality). Schizophrenic objects.
Part II. The dualities in Part I relate propositional logics (algebras) and their semantics (topological spaces). These dualities can be extended to modal logics and their Kripke semantics. The main idea is that modal logics are described by L-algebras where L is a functor on a category of algebras and Kripke models are described by T-coalgebras where T is a functor on the dual category of topological spaces.