Quantum uncertainty in a macroscopic domain

Abstract

We develop a classical model-theoretic representation of partial Boolean algebras and use it to formulate quantum-like uncertainty without abandoning classical propositional logic. Given a surjection from the initial formulæ of a propositional language onto a partial Boolean algebra $(\mathcal V,Π)$, we construct a consistent theory $\mathcal T_g$ whose core---the ordered set of equivalence classes of initial formulæ---is isomorphic to $(\mathcal V,Π)$. Its models are characterized as upward-closed clusters, yielding a model-theoretic formulation of KS-colourability: an $n$-dimensional partial Boolean algebra is KS-colourable exactly when the induced theory has a model meeting every pre-frame in one primitive formula. For finite-spectrum observables, uncertainty is defined by the number of locally admissible atomic outcomes. Dispersion-free models are characterized by the singleton pre-frame condition. With an additional measurement-update postulate, a finite example shows how measurement of an incompatible observable can destroy sharpness, providing a model-theoretic form of back-action. A $12$-vertex partial Boolean algebra is KS-colourable, whereas a rigorously constructed $140$-vertex, four-dimensional partial Boolean algebra is not, as shown by a parity argument; its incidence structure is isomorphic to the Peres $24$-ray, $24$-basis configuration in $\mathbb R^4$. Macroscopic interpretations show that these phenomena arise from the organization of propositions and models rather than from nonclassical deduction. Finally, we relate certain models to the probability-one propositions of pure states and density operators, while emphasizing that such certainty models do not determine the full quantum state.