Classical Logic and Quantum Logic with Multiple and 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...
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Wiley
2016-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2016/6830685 |
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| author | Mladen Pavičić |
| author_facet | Mladen Pavičić |
| author_sort | Mladen Pavičić |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-77da822bb0c14996a208fc5b94c538b8 |
| institution | OA Journals |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-77da822bb0c14996a208fc5b94c538b82025-08-20T02:07:09ZengWileyAdvances in Mathematical Physics1687-91201687-91392016-01-01201610.1155/2016/68306856830685Classical Logic and Quantum Logic with Multiple and Common Lattice ModelsMladen Pavičić0Department of Physics-Nanooptics, Faculty of Mathematics and Natural Sciences, Humboldt University of Berlin, Berlin, GermanyWe 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.http://dx.doi.org/10.1155/2016/6830685 |
| spellingShingle | Mladen Pavičić Classical Logic and Quantum Logic with Multiple and Common Lattice Models Advances in Mathematical Physics |
| title | Classical Logic and Quantum Logic with Multiple and Common Lattice Models |
| title_full | Classical Logic and Quantum Logic with Multiple and Common Lattice Models |
| title_fullStr | Classical Logic and Quantum Logic with Multiple and Common Lattice Models |
| title_full_unstemmed | Classical Logic and Quantum Logic with Multiple and Common Lattice Models |
| title_short | Classical Logic and Quantum Logic with Multiple and Common Lattice Models |
| title_sort | classical logic and quantum logic with multiple and common lattice models |
| url | http://dx.doi.org/10.1155/2016/6830685 |
| work_keys_str_mv | AT mladenpavicic classicallogicandquantumlogicwithmultipleandcommonlatticemodels |