Quantum Logic as the Logic of Contexts

2026-07-09

AI Summary

This work challenges the conventional view that treats classical logic as the foundational framework from which quantum logic diverges, arguing instead that this perspective entails an ontological misalignment. Within a finitely computable setting, the paper employs direct integral decomposition of orthomodular lattices to factor free orthomodular lattices into a direct product of contextual registers and Boolean content, thereby constructing a calculus over context–bit-vector pairs. The central contributions are threefold: it demonstrates that classical logic arises as the information-loss quotient of contextual logic under a forgetting operation; reveals a rigid duality among six context levels demarcated by commutativity; and proves that this forgetting operation induces a surjective lattice homomorphism whose quotient algebra is precisely the classical Boolean algebra, thereby reestablishing quantum logic’s foundational status.

Abstract

Quantum logic is usually presented as a non-classical departure from ordinary reasoning forced on us by quantum mechanics, with classical logic kept as the secure starting point. We argue for the opposite order of explanation in a finite and fully computable setting. The free orthomodular lattice on two generators has ninety-six elements, the direct product of a six-element non-distributive factor and a sixteen-element Boolean factor. Reading the first factor as a register of contexts and the second as Boolean content, we obtain a calculus whose elements are context--bit-vector pairs and whose operations act component by component. With this calculus we establish three results. First, we classify the six layers by commutativity, identifying the central kernel of context-neutral propositions together with a dual central layer in which all complementary contexts are present. Second, we show that orthocomplementation rearranges the layers exactly as the complementation of the small factor rearranges its elements, which makes the duality among the layers rigid rather than accidental. Third, we prove that the operation forgetting the context is a surjective homomorphism of orthocomplemented lattices whose quotient is the classical Boolean algebra, so that classical logic is a six-to-one, information-losing image of the contextual calculus.

Problem

• quantum logic
• context
• orthomodular lattice
• Boolean algebra
• classical logic

Innovation

• quantum logic
• orthomodular lattice
• contextuality
• Boolean quotient
• lattice homomorphism