Kochen--Specker Contextuality and the Tetralemmatic Calculus of Quantum Mechanics (The Formal Roots of the Unaskable in Quantum Measurement)

Abstract

he contextual character of quantum observables—formalized by the Kochen--Specker theorem and Bohr's complementarity—represents a fundamental rupture in classical binary logic. In interference regimes, such as the double-slit experiment, polar questions (e.g., ``Which path did the particle take?'') are operationally unaskable, yet standard logical frameworks fail to formalize this unaskability without collapsing into truth-value gaps or triviality. We resolve this by introducing a context-indexed formal logic that adjoins an explicit \emph{applicability predicate} to the valuation space. Drawing structural inspiration from the ancient \emph{catuṣkoṭi} (tetralemma), we map quantum operational states onto four distinct logical sectors: \emph{Thesis/Antithesis} (strong measurement and decoherence), \emph{Synthesis} (paraconsistent gluts realized via weak measurement), and \emph{Holothesis} (paracomplete gaps representing typed inapplicability). From this logical foundation, we generate a $*$-algebra equipped with a positive warrant functional. Through the Gelfand--Naimark--Segal (GNS) construction, we demonstrate that Hilbert-space structure—and the complex unit $i$ itself—emerge canonically from an order-$4$ logical symmetry rather than being presupposed. Contextuality is quantified via incompatibility graphs, with chromatic number bounding minimal classical embeddings. We establish a Noether-like correspondence wherein physical symmetry breaking dictates logical polarity, while symmetry preservation dictates logical transcendence (self-duality). This \emph{logic-first reconstruction} unifies Relational Quantum Mechanics (Rovelli) and Consistent Histories (Griffiths, Omnès, Gell-Mann, Hartle) as regime-relative logical stances within a pre-geometric ontology of \emph{interdependent origination} (contextual co-arising of observer and observable), where the observer's contextual frame and the system's observable reality are co-arising and non-separable. The framework yields testable deviations in many-body interference ($I_3 \neq 0$), providing an empirical pathway for detection.

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