Discrete Distinguishability as the Ontological Basis of Logic

This work proposes a discrete foundation of logic grounded in distinguishability and order. Instead of taking truth values as primitive, the framework introduces the act of distinction as the fundamental unit of logical structure. We formulate the axiom of discrete distinguishability: only discretely distinguishable states are ontologically admissible. Under this assumption, logical systems are reconstructed as structures over distinctions and their ordering. Logical operations (negation, conjunction, disjunction, implication) are not assumed as primitives, but arise from relations such as composition, compatibility, and precedence. Classical and non-classical logics—including constructive, graded, modal, and non-explosive systems—are shown to emerge as specific configurations of distinction structures. Continuous domains are reinterpreted as limit representations of discrete systems rather than ontological primitives. The framework integrates syntax, semantics, model theory, and computation within a unified formal system based on realizability. It further establishes minimality: the primitives of distinction and order are sufficient for constructing logical systems, and no additional primitives are required. This work provides a unified and realizability-consistent perspective on logic, aligning logical theory with physical and computational constraints.