Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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’.  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 (Fernandez Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here  show that S4C is complete for continuous functions on the rational numbers.
['Philip Kremer']
I am writing a research paper on a idea The modal logic of continuous functions on the rational numbers
[{"from": "human", "value": "I am writing a research paper on a idea The modal logic of continuous functions on the rational numbers"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let \\({{\\mathcal L}^{\\square\\circ}}\\) be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality \u25a1 and a temporal modality \u25e6, understood as \u2018next\u2019.  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 \u3008X, f\u3009, 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 (Fernandez Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here  show that S4C is complete for continuous functions on the rational numbers."}]