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We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H. [ABSTRACT FROM AUTHOR]

Let $${{\mathcal L}^{\square\circ}}$$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language $${{\mathcal L}^{\square\circ}}$$ by interpreting $${{\mathcal L}^{\square\circ}}$$ in dynamic topological systems, i.e., ordered pairs 〈 X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. S4C is also complete for continuous functions on Cantor space (Mints and Zhang, Kremer), and on the real plane (Fernández Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here we show that S4C is complete for continuous functions on the rational numbers. [ABSTRACT FROM AUTHOR]

Let $$\mathcal{L}$$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality $$\square$$ and a temporal modality $$\bigcirc$$ , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language $$\mathcal{L}$$ by interpreting $$\mathcal{L}$$ in dynamic topological systems, i.e. ordered pairs $$\langle X, f\rangle$$ , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result. [ABSTRACT FROM AUTHOR]