Classical and Quantum Logics with Multiple and a Common Lattice Models

TL;DR

This paper explores multiple lattice models for classical and quantum logics, revealing a shared non-orthomodular lattice that allows the same logic to operate on both digital and non-digital computers, with implications for computational efficiency.

Contribution

It demonstrates the existence of a common non-orthomodular lattice model for both classical and quantum logics, enabling unified logical operations across different computational paradigms.

Findings

Classical logic has disjoint distributive and non-distributive ortholattice models.

Quantum logic has multiple disjoint lattice models, with only one orthomodular.

A shared non-orthomodular lattice model supports both classical and quantum logics.

Abstract

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and non-distributive ortholattices as its models. In particular, we prove that both classical and quantum logics are sound and complete with respect to each of these lattices. We also show that there is one common non-orthomodular lattice that is a model of both quantum and classical logics. In technical terms, that enables us to run the same classical logic on both a digital (standard, two subset, 0-1 bit) computer and on a non-digital (say, a six subset) computer (with appropriate chips and circuits). With quantum logic, the same six element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.