A Topological Semantics of Dialogue: Nerve Structures and Logical Extraction

We introduce a concise, topologically-motivated semantics for finite dialogues by mapping each utterance to an open set in a fixed semantic space, building the corresponding nerve complex of joint satisfiability, and extracting fundamental combinatorial invariants: 1. The negative nerve, which enumerates all finite collections of utterances whose opens have empty intersection, providing a straightforward criterion for merging separate transcripts without contradiction. 2. The global interpretation subspace, the unique minimal open in which all asserted utterances hold simultaneously, enabling effective enumeration of all logical consequences of the entire dialogue. 3. A practical demonstration in the Wolfram Language, with algorithms for constructing nerves, detecting inconsistencies, and computing the global interpretation, thereby illustrating computational feasibility. Our framework is grounded in classical duality and topological semantics (Stone duality, Priestley duality, Tarski's semantics, coherence-space methods, Scott domains, topos semantics, and homotopy type theory) while drawing on recent advances in topological data analysis and dialogue-based semantics.