Generalised Geometric Logic: A Logic for Expressing Neural Network Architectures

Neural networks achieve strong empirical performance, yet their architectural semantics and compositional structure remain difficult to analyse formally. This paper develops a logical–topological framework for reasoning about neural network architectures independently of learning dynamics. Focusing on the ART/LAPART family as a canonical testbed in which bidirectional interaction and stability are ex-plicit, we provide a semantic interpretation of excitation relations using geometric logic and topological systems. We extend the classical setting to fuzzy and frame-valued semantics in order to capture graded and potentially incomparable activation strengths. With this extension we show that we express SHAP - a Neural Network Explainability methodology. The contribution is foundational: it clarifies how ar-chitectural causal structure can be represented, compared, and composed. While the technical development centres on ART and LAPART, the framework isolates structural principles—compositionality, graded influence, and continuity—that can be extend to modern deep neural network architectures.