Quantum logic is typically presented as a non-classical mode of reasoning imposed by quantum mechanics, with classical logic serving as a secure starting point. We argue for an opposite order of explanation in a finite and fully computable setting. The free orthomodular lattice on two generators has ninety-six elements, being the direct product of a six-element non-distributive factor and a sixteen-element Boolean factor. Interpreting 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, with operations acting component-wise.
Using this calculus, we establish three results. First, we classify the six layers by commutativity, identifying the central kernel of context-neutral propositions along with a dual central layer containing all complementary contexts. Second, we show that orthocomplementation rearranges the layers exactly as the complementation of the small factor rearranges its elements, making the duality among the layers rigid rather than accidental. Third, we prove that the operation of forgetting the context is a surjective homomorphism of orthocomplemented lattices whose quotient is the classical Boolean algebra, indicating that classical logic is a six-to-one, information-losing image of the contextual calculus.
Blogger's Review: This article offers a fresh perspective on quantum logic, emphasizing the significance of context and rigorously demonstrating the profound relationship between context and classical logic. This approach not only enriches the theoretical framework of quantum logic but also opens up new avenues for future research.