Determinacy as a Two-Place Relation: The Category Mistake in Quantum Interpretation, and the Dissolution of the Paradox Set

A realist about quantum mechanics wants two things the standard interpretations will not sell together: enough counterfactual structure to say what an unperformed experiment would have shown, and no hidden valuation for the Bell theorems to kill. This paper argues that both can be had, and that having them is a forced consequence of one correction rather than a lucky feature of a formalism. Following Spekkens’s diagnosis that quantum interpretation is stalled on a category mistake — an early wrong entry propagated through a century of internally coherent work — the paper identifies the entry: determinacy has been treated as a one-place property that facts have or lack, when it is a two-place relation between an attribute and an information state. The incumbent interpretations are read as four strategies for protecting the one-place entry, and their failure to converge as the expected sociology of a grid worked from a wrong square. Two results carry the paper. The first is a no-go: the two-place structure cannot be deflated back into determinate-but-unknown values. A trace-free deflation requires an overwrite semantics the framework forbids; a deflation denied overwrite must place its determinate values either inside a persisting state, where they contradict that state’s own stated content, or outside every state, where their image is a single-sample-space model of exactly the kind the Boole–Fine–Pitowsky correspondence shows the quantum statistics exclude. The correction is therefore a change of category, not of notation, and the classicist who declines it wholesale pays in contradiction or false statistics, while one who declines it selectively is handed a menu of priced doors. The second result is a yield: the same correction, fixed in advance and applied with nothing added per case, dissolves the standard paradox set — delayed choice, the cat, Wigner’s friend, EPR and its foliation worry, PBR, the proper/improper mixture — each dissolution graded, with Frauchiger–Renner marked as the contested case. The argument is conditional on a four-entry ledger stated at the outset, and it closes by naming the squares still blank.