On Dynamic Topological Logic of the Real Line

This article explores the topological interpretations of the modal language with two modalities-□, which is interpreted as the interior operation and ◯ ('next') which is interpreted as the pre-image operation for a continuous function. It is known that the □◯ logic S4C is complete with respect to topological interpretations in ℝn for n≥2, yet it is incomplete with respect to topological interpretations in ℝ. We focus on the logic L □◯(ℝ) of all the □◯ formulas that are sound with respect to topological interpretations in ℝ. In this article we present two formulas in L □◯(ℝ)-S4C, and prove that they are sound in ℝ and independent. We also establish that the previously known examples of formulas in L □◯(ℝ)-S4C are instances of a particular consequence of one of the two formulas presented.