Standard Logics Are Valuation-Nonmonotonic

TL;DR

This paper demonstrates that standard propositional logics are valuation-nonmonotonic, with their models and valuations changing significantly when additional conditions are imposed, impacting their applicability to quantum and classical computing.

Contribution

It proves the completeness of these logics for broader classes of non-orthomodular lattices and introduces a technique to separate valuations into disjoint lattice classes.

Findings

Standard logics are valuation-nonmonotonic.

Models change drastically with added conditions.

Valuations can be separated into disjoint lattice classes.

Abstract

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 paper 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.