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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 6830685, 12 pages
http://dx.doi.org/10.1155/2016/6830685
Research Article

Classical Logic and Quantum Logic with Multiple and Common Lattice Models

1Department of Physics-Nanooptics, Faculty of Mathematics and Natural Sciences, Humboldt University of Berlin, Berlin, Germany
2Center of Excellence for Advanced Materials and Sensing Devices (CEMS), Photonics and Quantum Optics Unit, Ruđer Bošković Institute, Zagreb, Croatia

Received 4 May 2016; Revised 7 July 2016; Accepted 25 July 2016

Academic Editor: Giorgio Kaniadakis

Copyright © 2016 Mladen Pavičić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 nondistributive ortholattices. In particular, we prove that both classical logic and quantum logic are sound and complete with respect to each of these lattices. We also show that there is one common nonorthomodular lattice that is a model of both quantum and classical logic. In technical terms, that enables us to run the same classical logic on both a digital (standard, two-subset, 0-1-bit) computer and a nondigital (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.