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We show that monadic intuitionistic quantifiers admit the following temporal interpretation: "always in the future" (for $\forall $) and "sometime in the past" (for $\exists $). It is well known that Prior's intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf S4}$ and provide a full and faithful translation of ${\sf MIPC}$ into ${\sf TS4}$. We compare this new translation of ${\sf MIPC}$ with the Gödel translation by showing that both ${\sf TS4}$ and ${\sf MS4}$ can be translated fully and faithfully into a tense extension of ${\sf MS4}$ , which we denote by ${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for ${\sf MS4.t}$ using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. [ABSTRACT FROM AUTHOR]

Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas. [ABSTRACT FROM AUTHOR]

We introduce the concept of a Zemanian logic above S4.3 and prove that an extension of S4.3 is the logic of a Tychonoff HED-space iff it is Zemanian. [ABSTRACT FROM AUTHOR]

It is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable. [ABSTRACT FROM AUTHOR]

We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive subframe and strongly cofinal subframe logics over wK4. We obtain axiomatizations of all four kinds of subframe logics over wK4. We conclude by giving a number of examples of different kinds of subframe logics over wK4. [ABSTRACT FROM PUBLISHER]

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (⋀,→) homomorphisms, (⋀,→, 0) homomorphisms, and (⋀,→⋁,) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev's subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev's theorem that each intermediate logic can be axiomatized by canonical formulas. [ABSTRACT FROM AUTHOR]