Lattices of intermediate and cylindric modal logics Nick Bezhanishvili Abtract: In this thesis we study classes of intermediate and cylindric modal logics. Intermediate logics are the logics that contain the intuitionistic propositional calculus IPC and are contained in the classical propositional calculus CPC. Cylindric modal logics are finite variable fragments of the classical first-order logic FOL. They are also closely related to n-dimensional products of the well-known modal logic S5. In this thesis we investigate: 1. The lattice of extensions of the intermediate logic RN of the Rieger-Nishimura ladder. 2. Lattices of two-dimensional cylindric modal logics. In particular, we study: (a) The lattice of normal extensions of the two-dimensional cylindric modal logic S5^2 (without the diagonal). (b) The lattice of normal extensions of the two-dimensional cylindric modal logic CML_2 (with the diagonal). Our methods are a mixture of algebraic, frame-theoretic and order-topological techniques. In Part I of the thesis we give an overview of Kripke, algebraic and general-frame semantics for intuitionistic logic and we study in detail the structure of finitely generated Heyting algebras and their dual descriptive frames. We also discuss what we call frame-based formulas. In particular, we look at the Jankov-de Jongh formulas, subframe formulas and cofinal subframe formulas and we construct a unified framework for these formulas. After that we investigate the logic RN of the Rieger-Nishimura ladder. The Rieger-Nishimura ladder is the dual frame of the one-generated free Heyting algebra described by Rieger and Nishimura. Its logic is the greatest 1-conservative extension of IPC. It was studied earlier by Kuznetsov, Gerciu and Kracht. We describe the finitely generated and finite descriptive frames of RN and provide a systematic analysis of its extensions. We also study a slightly weaker intermediate logic KG, introduced by Kuznetsov and Gerciu. KG is closely related to RN and plays an important role in our investigations. While studying extensions of KG and RN we introduce some general techniques. For example, we give a systematic method for constructing intermediate logics without the finite model property, we give a method for constructing infinite antichains of finite Kripke frames that implies the existence of a continuum of logics with and without the finite model property. We also introduce a gluing technique for proving the finite model property for large classes of logics. In particular, we show that every extension of RN has the finite model property. Finally, we give a criterion of local tabularity in extensions of RN and KG. In Part II of the thesis we investigate in detail lattices of two-dimensional cylindric modal logics. The lattice of extensions of one-dimensional cylindric modal logic, is very simple: it is an (\omega + 1)-chain. In contrast to this, the lattice of extensions of the three-dimensional cylindric modal logic is too complicated to describe. In this thesis we concentrate on two-dimensional cylindric modal logics. We consider two similarity types: two-dimensional cylindric modal logics with and without diagonal. Cylindric modal logic with the diagonal corresponds to the full two-variable fragment of FOL and the cylindric modal logic without the diagonal corresponds to the two-variable substitution-free fragment of FOL. Cylindric modal logic without the diagonal is the two-dimensional product of S5, which we denote by S5^2. It had been shown that the logic S5^2 is finitely axiomatizable, has the finite model property, is decidable, and has a NEXPTIME-complete satisfiability problem. We show that every proper normal extension of S5^2 is also finitely axiomatizable, has the finite model property, and is decidable. Moreover, we prove that in contrast to S5^2 itself, each of its proper normal extensions has an NP-complete satisfiability problem. We also show that the situation for cylindric modal logics with the diagonal is different. There are continuum many nonfinitely axiomatizable extensions of the cylindric modal logic CML_2. We leave it as an open problem whether all of them have the finite model property. Finally, we give a criterion of local tabularity for two-dimensional cylindric modal logics with and without diagonal and characterize the pre-tabular cylindric modal logics.