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We show that subspaces of the space $${\mathbb{Q}}$$ of rational numbers give rise to uncountably many d-logics over K4 without the finite model property. [ABSTRACT FROM AUTHOR]

We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R. [ABSTRACT FROM AUTHOR]

When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α, where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α, where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that $${X \in {\bf H}_{1}}$$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that $${X \in {\bf H}_{n}}$$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have $${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$$ , and for a limit ordinal α, we have $${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. [ABSTRACT FROM AUTHOR]

Let $${\beta(\mathbb{N})}$$ denote the Stone–Čech compactification of the set $${\mathbb{N}}$$ of natural numbers (with the discrete topology), and let $${\mathbb{N}^\ast}$$ denote the remainder $${\beta(\mathbb{N})-\mathbb{N}}$$. We show that, interpreting modal diamond as the closure in a topological space, the modal logic of $${\mathbb{N}^\ast}$$ is S4 and that the modal logic of $${\beta(\mathbb{N})}$$ is S4.1.2. [ABSTRACT FROM AUTHOR]