Formal Reconstruction of the Qal Wā-ḥōmer from Axiomatic and Topological Perspectives
Formal Reconstruction of the Qal Wā-ḥōmer from Axiomatic and Topological Perspectives

Summary/Abstract: Traditional logics, including Aristotelian, Stoic, Epicurean, Indian, and Buddhist systems, might be dismissed as lacking formal axiomatization and standard algebraic structures. However, these “logics” inherently possess rigorous doctrines of inference. To bridge this gap, this paper evaluates ancient logical systems through the lens of modern pure mathematics, specifically Univalent Foundations and Homotopy Type Theory. We introduce the concept of the logeme: a finite, consistent fragment of a logical theory closed under its specific inference rules. Unlike infinite axiomatic systems, a logeme captures the discrete, self-contained “molecules” of historical reasoning. By factoring the formulas of a logeme by logical equivalence, we construct a Lindenbaum-Tarski algebra that naturally forms a partially ordered set (poset). We then map these posets to simplicial complexes to analyze their geometric realization. Guided by the Univalence Axiom, we define two logemes as mathematically identical if and only if their corresponding posets share the same homotopy type. This topological framework is applied to the a fortiori (qal wāḥōmer / kaimutika-nyāya) argument across Talmudic, Christian, Hindu, and Confucian traditions. By translating ancient inference rules into topological spaces, we demonstrate that these historical phenomena can be formalized, compared, and validated as rigorous mathematical structures, revealing a universal, univalent foundation underlying diverse ancient epistemologies.

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