Journal of Logic and Computation

Journal of Logic and Computation Advance Access originally published online on June 26, 2008
Journal of Logic and Computation 2008 18(6):959-982; doi:10.1093/logcom/exn018

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Pavicic, M. Articles by Megill, N. D. Search for Related Content Social Bookmarking          What's this?

© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

Standard Logics Are Valuation-Nonmonotonic

Mladen Pavii

Physics Chair, Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia.
E-mail: pavicic{at}grad.hr

Norman D. Megill

Boston Information Group, 19 Locke Ln. Lexington, MA 02420, USA
E-mail: nm{at}alum.mit.edu

Received 12 October 2007.

It has recently been discovered that both quantum and classical propositional logics can be modelled by classes of non-orthomodular and thus non-distributive lattices that properly contain standard orthomodular and Boolean classes, respectively. In this article we prove that these logics are complete even for those classes of the former lattices from which the standard orthomodular lattices and Boolean algebras are excluded. We also show that neither quantum nor classical computers can be founded on the latter models. It follows that logics are valuation-nonmonotonic in the sense that their possible models (corresponding to their possible hardware implementations) and the valuations for them drastically change when we add new conditions to their defining conditions. These valuations can even be completely separated by putting them into disjoint lattice classes by a technique presented in the paper.

Keywords: Nonmonotonic logic; classical logic; quantum logic; non-distributive ortholattice; non-orthomodular ortholattice; weakly orthomodular lattice; Boolean algebra; weakly distributive lattice; artificial intelligence

References

1. Beran L. Orthomodular Lattices; Algebraic Approach. (1985) Dordrecht: D. Reidel.
2. Boole G. An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. (1854) London: Walton & Maberly. Repr. Dover, New York, 1958.
3. Hilbert D, Ackermann W. Principles of Mathematical Logic. (1950) New York: Chelsea.
4. Kalmbach G. Orthomodular Logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik (1974) 20:395–406.[CrossRef][Web of Science]
5. Kalmbach G. Orthomodular Lattices. (1983) London: Academic Press.
6. Leblanc H. Alternatives to standard first-order semantics. In: Handbook of Philosophical Logic—Gabbay D, Guenthner F, eds. (1983) Dordrecht: D. Reidel. 189–274. vol. I: Elements of Classical Logic.
7. Megill ND, Pavii M. Deduction, ordering, and operations in quantum logic. Foundations of Physics (2002) 32:357–378.[CrossRef][Web of Science]
8. Megill ND, Pavii M. Equivalences, identities, symmetric differences, and congruences in orthomodular lattices. International Journal of Theoretical Physics (2003) 42:2797–2805.[CrossRef][Web of Science]
9. Mittelstaedt P. Quantum Logic. Vol. 18. (1978) London: Synthese Library, Reidel.
10. McKay BD, Megill ND, Pavii M. Algorithms for Greechie Diagrams. International Journal of Theoretical Physics (2000) 39:2381–2406.[CrossRef][Web of Science]
11. Pavii M. Identity rule for classical and quantum theories. International Journal of Theoretical Physics (1998) 37:2099–2103.[CrossRef][Web of Science]
12. Pavii M. Minimal quantum logic with merged implications. International Journal of Theoretical Physics (1987) 26:845–852.[CrossRef][Web of Science]
13. Pavii M. Quantum Computation and Quantum Communication: Theory and Experiments. (2005) New York: Springer.
14. Pavii M, Megill ND. Binary orthologic with modus ponens is either orthomodular or distributive. Helvetica Physica Acta (1998) 71:610–628.[Web of Science]
15. Pavii M, Megill ND. Is quantum logic a logic? In Handbook of Quantum Logic and Quantum Structures, Vol. In: Quantum Logic—Engesser K, Gabbay D, Lehmann D, eds. (2008) Amsterdam: Elsevier. 31–54.
16. Pavii M, Megill ND. Non-orthomodular models for both standard quantum logic and standard classical logic: repercussions for quantum computers. Helvetica Physica Acta (1999) 72:189–210.[Web of Science]
17. Pavii M, Megill ND. Quantum Logic and Quantum Computation. In: In Handbook of Quantum Logic and Quantum Structures. Vol. Quantum Structures—Engesser K, Gabbay D, Lehmann D, eds. (2007) Amsterdam: Elsevier. 751–787.
18. Pavii M, Merlet J-P, Megill ND. Exhaustive enumeration of Kochen–Specker vector systems. In: The French National Institute for Research in Computer Science and Control Research Reports RR-5388 (2004).
19. Pavii M, Merlet J-P, McKay BD, Megill ND. Kochen–Specker Vectors. Journal of Physics A (2005) 38:497–503. Corrigendum, Journal of Physics A, 38, 3709, 2005.
20. Quine WV. Methods of Logic. (1970) 3rd edn. London: Routledge & Kegan Paul.
21. Schechter E. Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions. (2005) Princeton: Princeton University Press.

CiteULike    Connotea    Del.icio.us    What's this?

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Pavicic, M. Articles by Megill, N. D. Search for Related Content Social Bookmarking          What's this?