Quantum Logic as Classical Logic

TL;DR

This paper demonstrates that quantum logic can be represented within a classical modal logic framework, challenging the view that quantum logic is inherently non-classical and showing that quantum propositional inference is fundamentally classical with a modal extension.

Contribution

The authors provide a lattice-embedding of orthomodular lattices into Boolean algebras with a modal operator, showing quantum logic as a classical modal logic and refuting prior theses.

Findings

Quantum logic can be embedded into classical modal logic.

A single modal operator captures quantum observation subjectivity.

Quantum negation corresponds to classical negation of observability.

Abstract

We propose a semantic representation of the standard quantum logic QL within a classical, normal modal logic, and this via a lattice-embedding of orthomodular lattices into Boolean algebras with one modal operator. Thus our classical logic is a completion of the quantum logic QL. In other words, we refute Birkhoff and von Neumann's classic thesis that the logic (the formal character) of Quantum Mechanics would be non-classical as well as Putnam's thesis that quantum logic (of his kind) would be the correct logic for propositional inference in general. The propositional logic of Quantum Mechanics is modal but classical, and the correct logic for propositional inference need not have an extroverted quantum character. One normal necessity modality suffices to capture the subjectivity of observation in quantum experiments, and this thanks to its failure to distribute over classical disjunction. The key to our result is the translation of quantum negation as classical negation of observability.