Classical and Quantum Logics with Multiple and a Common Lattice Models

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.