Is Quantum Logic a Logic?

TL;DR

This paper demonstrates that quantum logic is fundamentally similar to classical logic, both being axiomatic systems with different models, challenging the notion that quantum logic is uniquely non-classical.

Contribution

It introduces novel lattice models for quantum and classical logics that are non-orthomodular and non-distributive, showing their equivalence as logical systems.

Findings

Quantum logic and classical logic are both axiomatic deductive systems.

Novel lattice models reveal that orthomodularity and distributivity are not intrinsic to quantum and classical logics.

Semantics depend on models; quantum and classical logics differ more in their models than in their axioms.

Abstract

It is shown that quantum logic is a logic in the very same way in which classical logic is a logic. Soundness and completeness of both quantum and classical logics have been proved for novel lattice models that are not orthomodular and therefore cannot be distributive either - as opposed to the standard lattice models that are orthomodular and distributive for the respective logics. Hence, we cannot attribute the orthomodularity to quantum logic itself, and we cannot attribute the distributivity to classical logic itself. The valuations of logics with respect of novel models turn out to be non-numerical, and therefore truth values and truth tables cannot in general be ascribed to the propositions of logics themselves but only to the variables of some of their models -for example, the two-valued Boolean algebra. Logics are, first of all, axiomatic deductive systems, and if we stop short of considering their semantics (models, valuations, etc.), then both quantum and classical logics will have a completely equal footing in the sense of being two deductive systems that differ from each other in a few axioms and nothing else. There is no ground for considering either of the two logics more "proper" than the other. Semantics of these logics belong to their models, and we show that there are bigger differences between the two aforementioned classical models than between two corresponding quantum and classical models.