Logemes as a new approach to metalogic

Abstract In this paper, we introduce the concept of logemes, a novel framework for reasoning that bridges nontrivial fragments of formal logic and topological spaces. Logemes are defined as logic diagrams that capture logical relationships relevant to specific datasets or problems. Unlike standard symbolic logic, which emphasizes completeness and formal validation within an algebraic framework, logemes focus on pragmatic, nontrivial subsets of inference rules tailored to real-world reasoning tasks. This approach builds on historical tools like logic diagrams (e.g. the square of opposition) while leveraging the combinatorial and topological richness of simplicial complexes. The paper explores the interplay between syntax and semantics through logemes, using topology to compare logical fragments across different systems. By focusing on structural properties rather than syntactic or semantic details, logemes reveal deeper similarities and differences between reasoning patterns, even when derived from distinct logical frameworks. The narrative alternates between bottom-up and top-down approaches: starting with basic components to build complex systems and simplifying complete systems into manageable fragments. Finally, logemes are connected to homotopy types, positioning them as a bridge between reasoning and topology. This unified framework offers a modular, flexible tool for studying logical fragments. By grounding reasoning in the topology of simplicial complexes, logemes provide a novel paradigm for analysing datasets and logical systems.