The Tangled Derivative Logic of the Real Line and Zero-Dimensional Space

In a topological setting in which the diamond modality is interpreted as the derivative (set of limit points) operator, we study a ‘tangled derivative’ connective that assigns to any finite set of propositions the largest set in which all those propositions are strictly dense. Building on earlier work of ourselves and others we axiomatise the resulting logic of the real line. We then show that the logic of any zero-dimensional dense-initself metric space is the ‘tangled’ extension of KD4, eliminating an assumption of separability in previous results for zero-dimensional spaces. This requires new kinds of ‘dissection lemma’ in the sense of McKinsey-Tarski. We extend the analysis to include the universal modality, and also show that the tangled extension of KD4 has a strong completeness result for topological models that fails for its Kripke semantics.