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
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Original Articles |
Standard Logics Are Valuation-Nonmonotonic
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Physics Chair, Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia.
E-mail: pavicic{at}grad.hr
Boston Information Group, 19 Locke Ln. Lexington, MA 02420, USA
E-mail: nm{at}alum.mit.edu
Received 12 October 2007.
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Abstract |
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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