Correspondence and canonicity in non-classical logic

In this thesis we study correspondence and canonicity for non-classical logic using algebraic and order-topological methods. Correspondence theory is aimed at answering the question of how precisely modal, first-order, second-order languages interact and overlap in their shared semantic environment. The line of research in correspondence theory which concerns the present thesis is Sahlqvist correspondence theory --- which was originally developed for classical modal logic, and provides a systematic translation between classical modal logic and first-order logic. Canonicity is closely related to correspondence, and ensures that logics axiomatized by these formulas are complete with respect to relational semantics. Thus, correspondence and canonicity together establish that Sahlqvist logics are semantically complete with respect to first-order definable classes of relational structures.
The first part of the thesis focuses on algebraic methods. In chapter 3, we prove the classical Sahlqvist correspondence theorem for basic modal logic in the algebraic setting of complex algebras of frames. We extend the algorithm ALBA to regular modal logic (modal logic with non-normal modalities) and intuitionistic modal mu-calculus in Chapters 4 and 5, respectively. In Chapter 6, we develop ALBA for distributive lattice expansions, using which we prove relativised canonicity for the meta-inductive inequalities. The second part of the thesis focuses on order-topological methods. In Chapter 7, we prove a modal-like duality for de Vries algebras. In Chapter 8, we prove a Sahlqvist correspondence and canonicity theorem for topological fixed-point logic on compact Hausdorff spaces.