Order-topological and model-theoretic methods for modal logics

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The research concerns the fundamental theory of modal logic and some other non-classical logics. Using model-theoretic, topological, algebraic, order-theoretic, and category-theoretic tools, general completeness theorems using enhanced Sahlqvist theory and canonical formulas will be proved, as well as specific completeness theorems for a number of logical systems. The research will also develop Fine's canonicity theorem and the theory of algebraic completions, and extend the theory of bisimulations to descriptive frames (Kripke frames endowed with additional topological structure). Descriptive frames will be an important tool, and the research will contribute to general understanding of their properties. New tools and techniques will be developed for dealing with them, and these will be of wider use. The research is part of a growing field that applies algebraic and topological methods in non-classical logics.