Standard Logics Are Valuation-Nonmonotonic

It has recently been discovered that both quantum and classical propositional logics can be modelled by classes of nonorthomodular 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.

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Mladen Pavicic, Norman D. Megill
• Keywords: Classical Propositional Logics, CORR 2008, Education, Non-distributive Lattices, Standard Orthomodular Lattices