Logemes and Their Homotopy-Theoretic Foundations

Abstract

We introduce logemes, consistent fragments of reasoning closed under at least one inference rule, as foundational units for different logics. Unlike full logical theories, logemes need not be axiomatized or algebraically structured; instead, they are evaluated via their associated Lindenbaum-Tarski quotients, interpreted as spaces presenting partially ordered sets. We propose that two logemes are homotopy identical precisely when their posetal semantics are homotopy equivalent. This criterion, grounded in the Univalence Axiom of Homotopy Type Theory, allows us to formalize diagrammatic reasoning and compare ancient logical traditions, such as Stoic and Yog¯ac¯ara, on purely mathematical grounds. We show that both traditions instantiate the same homotopy type of poset, confirming heir logical identity despite historical separation.